Microeconomics - some optional questions | Mastering Key Economic Concepts: Understanding Demand, Elasticity, and Utility

Q No: 1


Which of the following factors can cause a shift in the demand curve for a product?

a. Changes in the price of complementary goods

b. All of these - Right Answer

c. Changes in the availability of substitute goods

d. Changes in consumer preferences

Changes in consumer preferences, such as a shift in tastes or preferences towards or away from a product, can lead to a change in demand. Changes in the price of complementary goods, which are goods consumed together with the product in question, and changes in the availability of substitute goods can also impact the demand for the product, resulting in a shift of the demand curve.

Q No: 2

The price elasticity of demand for a good is said to be inelastic if:

a. The quantity demanded of the good is highly responsive to changes in its price.

b. The price of the good is not responsive to changes in its quantity demanded.

c. The price of the good is highly responsive to changes in the quantity demanded.

d. The quantity demanded of the good is not responsive to changes in its price.- Correct Option

The quantity demanded of the good is not responsive to changes in its price. When the price elasticity of demand for a good is inelastic, it means that the quantity demanded is relatively unresponsive or less sensitive to changes in the price of the good. Inelastic demand implies that even significant changes in price result in proportionally smaller changes in the quantity demanded. The demand for inelastic goods is less affected by price fluctuations.

Q No: 3

A restaurant lowered the price of a meal from $20 to $15 and observed that the number of meals sold increased from 100 to 120. What is the price elasticity of demand for the meal? Use the arc price elaticity formula.

a. -1.5

b. -0.64 - Correct Answer

c. -1.0

d. -0.5

The price elasticity of demand can be calculated using the arc price elasticity formula:

Elasticity = (Percentage change in quantity demanded) / (Percentage change in price)

Using the given values, the percentage change in quantity demanded is (120-100)/((100+120)/2) = 20/110 = 0.1818.

The percentage change in price is (15-20)/((20+15)/2) = -5/17.5 = -0.2857.

Plugging these values into the formula, we get -0.1818/-0.2857 = -0.6364, which can be rounded to -0.64. Therefore, the price elasticity of demand for the meal is -0.64.

Q No: 4

Which of the following statements best describes the concept of marginal utility?

a. Marginal utility measures the price elasticity of demand for a product.

b. Marginal utility measures the total satisfaction a consumer derives from a product.

c. Marginal utility measures the opportunity cost of consuming a product.

d. Marginal utility measures the additional satisfaction a consumer derives from consuming one more unit of a product. - Correct Answer

Marginal utility is the change in total utility resulting from a one-unit change in the consumption of a good or service. It is a measure of the additional satisfaction or usefulness that a consumer derives from consuming an additional unit of a product.

Q No: 5

If two products have a cross-price elasticity of demand of -1.2, what does that mean?

a. A 12% increase in the price of one product will cause a 10% decrease in the demand for the other product.

b. A 10% increase in the price of one product will cause a 12% decrease in the demand for the other product.- Correct Option

c. A 10% increase in the price of one product will cause a 10% decrease in the demand for the other product.

d. A 10% increase in the price of one product will cause a 12% increase in the demand for the other product.

A 10% increase in the price of one product will cause a 12% decrease in the demand for the other product. A negative cross-price elasticity of demand indicates that the two products are complements. In this case, a cross-price elasticity of -1.2 suggests that a 10% increase in the price of one product leads to a 12% decrease in the demand for the other product. The percentage change in demand is 1.2 times the percentage change in price, but with the opposite sign due to the negative elasticity.

Q No: 6

The law of demand assumes that:

a. Consumer income is variable

b. All other factors influencing demand remain constant - Correct Answer

c. There are no substitute goods available

d. The good is a necessity

The law of demand assumes the ceteris paribus condition, meaning "all other things being equal". This implies that while considering the effect of price on quantity demanded, all other influencing factors are held constant.

Q No: 7

Suppose a consumer has a fixed income and can only buy two products: apples and oranges. The price of apples is $1 per pound and the price of oranges is $2 per pound. The consumer's marginal utility per pound of apples is 10 and their marginal utility per pound of oranges is 20. To achieve consumer equilibrium, what should the consumer do?

a. The consumer is already at equilibrium, as the marginal utility per dollar spent is the same for each product. - Correct Answer

b. Buy only as many oranges as they can afford, since oranges have a higher price and provide more total utility.

c. Buy only oranges, since they provide more utility per dollar spent.

d. Buy only apples, since they have a lower price and provide more utility per dollar spent.

To determine consumer equilibrium, we compare the marginal utility per dollar of each product. In this case, the marginal utility per dollar spent on both apples and oranges is 10 (10/1 = 10 and 20/2 = 10). Since the marginal utility per dollar is equal for both products, the consumer is already maximizing their utility given the prices and their fixed income. Therefore, there is no need for the consumer to change their purchasing behavior.

Q No: 8

If a consumer's income decreases and all other factors remain constant, the consumer's DEMAND for an inferior good will:

a. Increase - Correct Answer

b. Decrease

c. Remains unchanged

d. Depend on the elasticity of demand for the good.

When a consumer's income decreases, the demand for an inferior good tends to increase. Inferior goods are those for which demand decreases as income increases. Conversely, as income decreases, consumers may opt for lower-cost alternatives, leading to an increase in the demand for inferior goods. Therefore, a decrease in income would result in an increase in the consumer's demand for an inferior good, assuming all other factors remain constant.

Q No: 9

If a product has an income elasticity of demand of 1.5, what does that mean?

a. A 10% increase in income will cause a 5% increase in demand.

b. A 15% increase in income will cause a 10% increase in demand.

c. A 10% increase in income will cause a 15% increase in demand. - Correct Answer

d. A 10% increase in income will cause a 10% increase in demand.

A 10% increase in income will cause a 15% increase in demand. A positive income elasticity of demand indicates a normal good, where as income increases, the quantity demanded also increases. In this case, income elasticity of demand of 1.5 suggests that a 10% increase in income will lead to a 15% increase in the demand for the product. The percentage change in demand is 1.5 times the percentage change in income.

Q No: 10

Which of the following is an example of the law of diminishing marginal utility in action?

a. A consumer buys three pizzas and enjoys each one equally.

b. A consumer buys three pizzas and enjoys each one more than the previous one.

c. A consumer buys three pizzas and enjoys the first one the most, the second one less, and the third one even less. - Correct Answer

d. A consumer buys three pizzas and enjoys the third one the most, the second one less, and the first one even less.

The above scenario demonstrates the law of diminishing marginal utility, which states that as a consumer consumes more of a particular good, the additional satisfaction or utility derived from each additional unit of the good diminishes. In this case, the consumer's enjoyment of the pizzas decreases with each subsequent pizza consumed, reflecting the diminishing marginal utility.

Q No: 11

What is meant by diseconomies of scale in the long run cost analysis?

a. A situation where long run average costs decrease as output increases.

b. A situation where long run average costs fluctuate as output increases.

c. A situation where long run average costs increase as output increases. - Correct Option

d. A situation where long run average costs remain constant as output increases.

Diseconomies of scale refer to a situation where long run average costs increase as output increases. This happens when increasing the scale of production leads to inefficiencies that increase the cost per unit of output. Refer to Theory of Cost. 

Q No: 12

If a firm is operating in stage III of the production process, it should ____ ?

a. Reduce the amount of the variable input being used. - Correct Option

b. Keep the amount of the variable input constant.

c. Continue to add more of the variable input.

d. Increase the amount of fixed inputs.

In stage III, the marginal product of the variable input is negative, which means that adding more of this input actually reduces total output. Therefore, the firm should reduce the amount of the variable input being used in order to increase its total output and move back into stage II.

Refer to 3 states of production

Q No: 13

What does the Marginal Rate of Technical Substitution (MRTS) represent?

a. The rate at which a firm can increase output by increasing one input while holding all other inputs constant.

b. The rate at which a firm's total costs increase as output increases.

c. The rate at which a firm's total revenue increases as output increases.

d. The rate at which a firm can substitute capital for labor without affecting output. Correct Option

The MRTS is the rate at which a firm can substitute one input for another (like labor for capital) without affecting the level of output. It is the slope of the isoquant at a given point.

Refer to topic Isoquants.

Q No: 14

What does the slope of an Isoquant represent?

a. The ratio of the prices of the two inputs used in production.

b. The rate at which one input can be substituted for another without changing the output. - Correct Option

c. The rate at which output changes as one input is increased and the other is held constant.

d. The ratio of the quantities of the two inputs used in production.

The slope of an isoquant represents the Marginal Rate of Technical Substitution (MRTS), which is the rate at which one input can be substituted for another without changing the output.

Refer to topic Isoquants.

Q No: 15

What does a cost function represent in microeconomics?

a. The relationship between the quantity of inputs used and the quantity of output produced.

b. The relationship between the price of a good and the quantity demanded.

c. The relationship between the output and the cost incurred in production of that output. - Correct Option

d. The relationship between the price of a good and the quantity supplied.

A cost function represents the relationship between the quantity of output produced and the cost of production. It shows how costs change as output levels change, holding other factors constant.

Refer to Theory of Cost.

Q No: 16

What does Marginal Product (MP) signify?

a. The ratio of total output to the number of inputs.

b. The average output produced per unit of input.

c. The total output produced per unit of input.

d The additional output produced by adding one more unit of input. - Correct Option

The Marginal Product (MP) of an input is the additional quantity of output that is produced when one additional unit of that input is used, holding all other inputs constant.

Refer to Production concepts.

Q No: 17

What does the slope of an Isocost line represent?

a. The ratio of the quantities of the two inputs used in production.

b. The ratio of the prices of the two inputs used in production. - Correct Option

c. The rate at which output changes as one input is increased and the other is held constant.

d. The rate at which one input can be substituted for another without changing the output.

The slope of an Isocost line represents the ratio of the prices of the two inputs used in production. This is because each point on the Isocost line represents a combination of inputs that the firm can afford given its budget, so the slope of the line indicates how much of one input the firm must give up to afford an additional unit of the other input.

Q No: 18

Which of the following statements is true about economies of scale?

a. Economies of scale occur when long run average costs decrease as output increases. - Correct Option

b. Economies of scale occur when long run average costs are independent of output.

c. Economies of scale occur when long run average costs remain constant as output increases.

d. Economies of scale occur when long run average costs increase as output increases.

Economies of scale refer to the cost advantages gained by increasing the scale of production. As output increases, economies of scale allow for spreading fixed costs over a larger quantity of output, leading to a decrease in long run average costs. This is due to factors such as specialization, bulk purchasing, and efficient use of resources, resulting in cost efficiencies and lower average costs per unit of output.

Refer to the Theory of Cost.

Q No: 19

What is the concept of Output Elasticity? Assume labor is the only variable input.

a. It measures how much total output will change if labor changes by a certain percentage. - Correct Option

b. It measures how much the average product will change if labor changes by one unit.

c. It measures how much total output will change if labor changes by one unit.

d. It measures how much the marginal product will change if labor changes by one unit.

Output Elasticity is a measure of the responsiveness of output to a percentage change in labor. It is used to assess the impact of a change in inputs on the level of output.

Refer to the Production concept topics.

Q No: 20

A firm employs 50 units of labor at a wage rate of $20 per unit, and 100 units of capital at a rental rate of $10 per unit, what is the firm's total cost of production?

a. $2000 - Correct Option

b. $4000

c. $5000

d. $3000

The firm's total cost of production can be calculated as C = wL + rK. In this case, w (the wage rate) is $20, L (the quantity of labor) is 50, r (the rental rate of capital) is $10, and K (the quantity of capital) is 100. Therefore, the total cost is $20*50 + $10*100 = $1000 + $1000 = $2000.

प्रसंग - आदर्श तिवारी | Prasang - Adarsh Tiwary

प्रसंग

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लफ़्ज़ों में इज़हार करना चाह रहा था जिन बातों को, I wanted to express in words those things,

उस जज़्बात के मायने मेरे लिए बड़े ख़ास थे, The meanings of those emotions were very special to me,

इशारे काफी थे समझाने को, Gestures were enough to explain,

पर सुनने वो लफ्ज़ तुम वही मेरे साथ थे। But to hear those words, you were right there with me.

धीमी धीमी बूंदों की बारिश, The rain of slow droplets,

बरश रही थी नन्ही बूंदे Tiny droplets were falling,

कर रही थी श्रृंगार तेरा, Adorning you,

मृगनैनी पलकों के तेरे । Your doe-like eyelashes.

चिन्हित करे सौंदर्य को ऐसी सकशियत तेरी, Such is your personality that marks beauty,

मुख दर्शाती स्वप्नसुंदरी,रंग तेरा सुनेहरा Your face shows a dream beauty, your color golden

कमल पंखुड़ी से पवित्र तू, मन चुलबुल अति चंचल Pure as a lotus petal, your mind playful and very fickle,

हर बात में तेरी सोच सही, सलाह भाव अति गहरा । In every matter, your thoughts correct, your advice deeply felt.

कैसे मैं पहल, नू शब्दों के जाल संग तेरे, How do I initiate, with the net of words along with you,

जानता हु तेरे दिल का राज़ पर कैसे निकालू तुमसे वो बात I know the secret of your heart but how to bring up that matter with you

तेरे भी नज़रिए में अपने को तौलना बड़ा ज़रूरी था, It was very important to weigh myself in your perspective too,

तुमने जो संकेत दिए, वो भाव नित नृत्य मयूरी था । The signs you gave, those feelings were like a dance of the peacock.

स्वाभाविक विचार सामान,सम्मान था मेरे प्यार को Natural thoughts were common, respect was there for my love

अत्यंत मतभेद के पार, मुझे मेरा नया संसार स्वीकार था ।। Beyond great differences, I had accepted my new world.

- आदर्श तिवारी -

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चित्रण - आदर्श तिवारी | Chitran - Adarsh Tiwary

चित्रण ____________________________

चौकोर सफ़ेद पन्ने सा जीवन चरित्र था मेरा My life was like a square white page.

कोई दाग नहीं, कुछ भी नहीं, अंत किनारा गहरा

No stains, nothing at all, just a deep edge.

जब बरस पड़े कुछ रंग, लाल हरे नीले से सारे

When some colors fell, all red, green, blue,

चित्र बना वो ऐसा, सुनहरे चमक मे इंद्रधनुष उखर पड़ा हो जैसे

The picture was such, as if a rainbow had emerged in golden radiance.

कागज के कश्ती बन, दरिया में मैं बह चला

Becoming a paper boat, I flowed into the river.

कश्ती और पानी का, शुरू हुआ यूं सिलसिला

Thus began the sequence of the boat and the water.

थपेड़ो को झेलता, तूफानों से खेलता 

Enduring the slaps, playing with the storms,

आगे बढ़ चला मैं काफिर सा फुहारो संग संभालता

I moved forward, managing like an infidel with the showers.

तूफानो के आगाज से कायर सदैव घबराते हैं,

Cowards always get scared from the onset of storms,

रंग भरी इस दुनिया में, कोरे ही रह जाते हैं

In this colorful world, they remain blank.

जो सैलाब न आते तो ठहराव का जिक्र न होता

Had the flood not come, there wouldn't be mention of stagnation.

जो फुहारे समझ इन्हे पार न करते, उन्हे वीर कहा ना जाता

Those who understand these showers and do not cross, they were not called brave.

रास्ते आसान सभी को भाते है,

Everyone likes easy paths,

कुछ अपने रास्ते खुद लिख जाते है

Some write their own paths.

जोखिम उठाना हर एक के बस का नहीं,

Taking risks is not everyone's cup of tea,

चुने कल को गढ़ने के अपने मायने होते हैं

Choosing to shape tomorrow has its own meanings.

रास्ता नया हर वक़्त सहारे को पुकारा करती है

The new path always calls for support.

हर शाम ढलने पर निशा सूरज को आवाज़ देती हैं

Every evening, the night calls out to the sun.

आत्म परिभाषित ये जीवन का मेरा यह एक अंश हैं

Self-defined, this is a part of my life.

प्रीत संकल्प का चित्रण है, यह मेरे स्वेत लेखन मे ||

It is the portrayal of the resolution of love, in my white writings.

- आदर्श तिवारी -

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यादों भरी वो लाइब्ररी - आदर्श तिवारी | Yaadon Bhari wo Library - Adarsh Tiwary

यादों भरी वो लाइब्ररी 

_____________________

 आज दिखी वो लाइब्ररी जहा हम तुम जाया करते थे,
Today I saw that library where we used to go,

ज़िन्दगी के पन्नो में बातें उखेरा करते थे|
We used to etch our conversations into the pages of life.
आज उन यादो से जुड़ने का एक और द्वार मिल गया,
Today I found another gateway to connect with those memories,

और मिला वो आधा प्यार, जो कभी यादों में था दब गया|
And found that half-love which had once been buried in memories.

किताबों से भरी हर वो तख़्त निहारा करती थी,
I used to gaze at every shelf filled with books,

जब तुम  उस कोने से मुझे पुकारा करती थी|
When you used to call out to me from that corner.

सन्नाटे मे हसने का तुम्हारा,वो एहसास मुझे फिर मिल गया, 
The feeling of your laughter in the silence, I found it again,

हर उस नज़ारे का मुझे,फिर से नज़ारा मिल गया|
I found that view of every sight once more.

  याद आई वो मेज़ जहा हम तुम संग बैठा करते थे,
I remembered that table where we used to sit together,

मिलती थी हमारी नज़रे ,घंटो बातें किया करते थे|
Our eyes would meet, and we would talk for hours.

वहा जाना तो बस, जाना पहचाना सा एक बहाना था,
Going there was just a familiar excuse,

वक्त की खाई मे भी,तुमसे मिलने जो आना था|
To meet you, even across the chasms of time.

रोज़ नए अध्याय हम अपने किस्से में जोड़ा करते थे,
Every day, we would add new chapters to our story,  

 तुम कहती थी, मैं सुनता था और ये दिन बीता करते थे|
You would speak, I would listen, and the days would pass.

 किताबों के पन्नो सा,जहां हम एक दूजे को पढ़ा करते थे,
Like the pages of a book, where we would read each other,

याद आ गयी वो लाइब्ररी जहा हम तुम जया करते थे। 
I remembered that library where we used to go.

 

~~~ आदर्श तिवारी ~~~

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Statistical Methods in Decision Making - Exploring the Relationship Between Height and Weight: A Comprehensive Analysis

 Questions:

1. How can we check if X = height and Y=weight are linearly related? 
2. What is the equation of the best fit line? 
3. Explain how we can test if the slope and intercept are significant? 
4. What does the T Test signify in regression?
5. Plot the residuals and explain your findings.
6. What are the assumptions of regression?
7. Predict the value of Y when X = 25 inches
   
   

Data Set:

Height(inches) / Weight(Lbs)

53 140.5
54 143
54 156
54 144
54 142
55 162.5
56 162
57 166.5
57 143
58 165
59 157.5
59 161.5
60 170
61 173.5
62 161
64 166
65 138
65 174.5
67 195
67 181.5
67 184
67 173.5
68 240
68 176
70 135
71 200.5
73 145
73 196.5
73 162
74 210

Q1: How can we check if X = height and Y=weight is linearly related?

Answer:
Visualizing the Data through scatter plot: 

Referring Excel sheet for regression statistics, we observe value for R2 = 0.28

Conclusion: Determining the strength and direction of the linear relationship between the Height and Weight based on the correlation coefficient value (r = 0.5254), we can simplify the interpretation as below: The correlation coefficient (r) ranges from -1 to +1. Since the value of r is positive (0.5254), it indicates a positive direction for the linear relationship. In other words, as height increases, weight also tends to increase. The value of r (0.5254) suggests that the linear relationship between height and weight is neither weak nor strong. It falls in the middle, indicating a moderate correlation. In conclusion, Height and Weight have a moderately positive linear relationship. As height increases, weight generally increases.

Q2: What is the equation of the best fit line?

Answer:
To find the equation of the best fit line for the data, we use linear regression, It helps us find a line that comes closest to all the points.
The equation is in the form

 y = mx + b.

➔ Y is the predicted weight
➔ X is the height, 'm' is the slope (how weight changes with height), and 'b' is the y-intercept (weight when height is zero).
➔ The regression model calculates m and b using the data points to get the best fit line.

This makes the above equation:
 m (value of slope) = 1.84
 b (y-intercept) = 51.902

y = 1.84x + 51.902

Q3: Explain how we can test if the slope and intercept are significant?

Answer: We can test the significance of the slope and intercept in a linear regression model using a t-test. The t-test helps determine if the estimated coefficients are significantly different from zero, indicating their statistical significance.
Data from Excel estimating the slope (m) and intercept (b) values

m (slope) = 1.84

b (y-intercept) = 51.902

p value = 0.0029 

For Slope: Null hypothesis,

H₀ = The slope of the regression line for Height is zero. 

Alternate hypothesis, 

H1 = The slope of the regression line for Height is not zero and bears a significant relation between height and weight.

Conclusion from data:
The p-value of 0.0029 indicates that there is strong evidence to reject the null hypothesis, which states that there is no significant relationship` between Height (inches) and Weight (lbs). This means that the relationship between height and weight is likely to be statistically significant. Therefore, based on this result, we can conclude that there is a meaningful relationship between height and weight.

For Intercept: Null hypothesis, H₀ = The intercept is not statistically significant, meaning that it does not have a meaningful impact on the relationship between the variables.

Alternate hypothesis, H1 = The intercept is statistically significant, suggesting that it does have a meaningful impact on the relationship between the variables.

Conclusion from data:
Based on the results obtained, the p-value for testing the significance of the intercept is 0.155. Since this p-value is higher than the usual significance level of 0.05, We cannot reject the null hypothesis. This means that there is not enough evidence to suggest that the intercept is statistically significant. In simpler terms, the intercept does not seem to have a significant impact on the relationship between the variables based on the available data. 

Q4: What does the T Test signify in regression?

Answer: Observing T test in regression;

Stating the Hypothesis:
The average expected increase in weight for each inch increase in height is estimated to be 1.84 lbs.
Null hypothesis (H₀): The slope for height is equal to zero (m = 0).
Alternative hypothesis (H₁): The slope for height is not equal to zero (m ≠ 0).

Determining the significance level: α @ 0.05, representing a 95% confidence interval.

Calculating the t-value by dividing the coefficient (1.84) by the standard error (0.563).
The calculated t-value is 3.268.

Conclusion based on the p-value:
The corresponding p-value for the calculated t-value (3.268) with degrees of freedom (df) = 28 is 0.00286751.
Since the p-value (0.00286751) is less than the chosen significance level (0.05),
we reject the null hypothesis (H₀: m = 0).

Therefore, we could state as below:

Based on the T-test, we conclude that height is statistically significant in the model. The relationship between height and weight is strong and Height significantly contributes to the accuracy of the model.


Q5: Plot the residuals and explain your findings.

Answer:

Referred value:
m (slope) = 1.840027
b (y intercept) 51.90161 

Predicted value = mx+b = 1.84*x+51.9 

Residual = observed value – predicted value for y

Based on same, tabled the values as below:

Below is the Scatter plot based on above residual values vs height.:

The table above displays the residuals, which indicate the differences between the predicted values and the observed values or the regression line. By examining the table, we can make the following observations:

1. For values between 150 and 170, the residuals tend to fall within the range of -10 to 10 for most of the values. This suggests that the model is fairly accurate and the predicted values align closely with the observed values or the regression line within this range.

2. However, for values greater than 170, the range of the residuals starts to increase. This indicates that the model becomes less linear and less accurate as the values increase. The relationship between the variables may not follow a straight-line pattern as strongly in this higher range.

3. On the other hand, in the lower ranges, the residuals remain relatively stable and show a somewhat linear relationship. This suggests that the model performs better and maintains a closer alignment between the predicted and observed values within these lower ranges.

4. In summary, the analysis of the residuals indicates that the model's accuracy and linearity vary depending on the value range. It appears to be more stable and linearly related in the lower ranges, but less accurate and less linear as the values increase. 

From scatter plot, its conclusive: As the values increase, the residuals also increase, indicating a weaker linear relationship between the variables. This is evident from the residual plot, which shows the scattered pattern of the residuals rather than a clear and consistent linear pattern.

Q6: What are the assumptions of regression?

Answer:

Simple Linear Relationship: A simple linear relationship implies a direct association between two variables that can be represented by a straight line on a scatter plot. It follows the equation: Y = mX + b, where Y is the dependent variable (e.g., weight), X is the independent variable (e.g., height), m is the slope of the line (indicating the rate of change of Y with respect to X), and b is the y-intercept (the value of Y when X is zero). This assumption suggests that the relationship between the variables can be adequately captured by a linear equation, such as a regression line.

Correlation between Variables: The correlation coefficient (r) measures the strength and direction of the relationship between two variables. In the context of height and weight, the correlation coefficient ranges from -1 to +1. A positive correlation coefficient suggests a positive linear relationship, indicating that taller individuals tend to have higher weights. A negative correlation coefficient would imply the opposite.

The correlation coefficient can be calculated using the formula:

r = (Σ((X-X̄)(Y-Ȳ)) / √(Σ(X-X̄)² * Σ(Y-Ȳ)²)

where X and Y represent the data points and X̄and Ȳ denote their respective means.

Variables are Independent:
The assumption of independence implies that the observations or data points in the dataset are not influenced by each other. In the case of height and weight, it means that the weight of one individual should not be affected by the height or weight of another individual. Each observation should represent a unique individual, and their heights and weights should not be influenced by external factors or the values of other individuals in the dataset.

Normally Distributed:
The assumption of normal distribution pertains to the shape of the data. It assumes that the distribution of the data points follows a bell-shaped curve, with most observations concentrated around the mean, and fewer observations in the tails. This assumption is related to the residuals in regression analysis, which should follow a normal distribution. The residuals can be obtained by subtracting the predicted values from the observed values. The normal distribution assumption allows for the application of statistical tests and enables accurate estimation and inference from the regression model.

By considering these assumptions, including a simple linear relationship, correlation between variables, independence of observations, and normal distribution, we can establish the foundation for reliable regression analysis. These assumptions, along with the associated mathematical formulas, help ensure the validity and accuracy of the regression model.

Q7: Predict the value of Y when X = 25 inches?

Answer:

m (slope) = 1.84 b (y-intercept) = 51.902
Standard error = 20.58
y = m+ bx ± standard error

•  Y = 51.902 + 1.84x ± 20.58 
• Since value of x = 25 
• Y= 51.902 + 46 ± 20.58
  Y= 97.902 ± 20.58
  i.e. Value of Y should be between 118.482 and 77.322
  
  
  Linear Regression Analysis#Height-Weight Correlation#Statistical Significance Testing#Regression Assumptions#Predictive Modelling#Data Visualization#Residual Analysis#T-Test in Regression#Best Fit Line Equation#Correlation Coefficients#Statistical Methods#Decision Making in Statistics#Predictive Analytics#Scatter Plot Analysis

Statistical Analysis Techniques: From Mean and Standard Deviation to Hypothesis Testing in Business Scenarios

Questions:

1. Compute the mean and standard deviation for below:

a) Sample data: 1,2 and 3

b) Population data: 10,20 and 30

2. A researcher conducted an experiment with a sample size of 36. The sample standard deviation was found to be 12. What is the standard error in this case?

3. There are two stocks X and Y. You have invested 50% of your wealth in each of these stocks. The covariance of the two stocks is 0.1. The variance of stock X is 0.1 and that of stock Y is 0.2. What is the correlation?

4. Find a symmetrically distributed interval around µ that will include 95% of the sample means when µ = 368, σ = 15, and n = 25.

5. You are interested in the average response time of customer support calls at a telecommunications company. You randomly select a sample of 100 customer support calls and find that the mean response time is 8 minutes, with a standard deviation of 3 minutes. Calculate a 90% confidence interval for the average response time of customer support calls at the company.

6. Is the average number of sales per month for a new sales strategy significantly different from the average number of sales per month for the previous sales strategy? (Consider alpha = 0.05)

New Sales Strategy Previous Sales Strategy

50 45

55 48

60 50

65 52

70 55

7. A company wants to determine if the average salary of its employees has increased over the past 10 years, taking into account both full-time and part-time employees. They take a random sample of 50 employees, 25 full-time and 25 part-time, to see whether there is a significant change from the average salary of $50,000 ten years ago. Perform Hypothesis testing (one sample t-test) to come to a conclusion. (Consider alpha = 0.05)

Refer: 

Dataset 

Employees salary (in $)

80,000

70,000

77,000

75,000

72,000

76,000

79,000

73,000

81,000

78,000

82,000

71,000

74,000

85,000

83,000

69,000

72,000

75,000

68,000

72,000

82,000

79,000

73,000

81,000

78,000

29,000

32,000

33,000

30,000

27,000

25,000

31,000

28,000

26,000

29,000

30,000

24,000

28,000

26,000

31,000

32,000

29,000

25,000

27,000

33,000

33,000

24,000

28,000

26,000

31,000

Answer: 

Q1. Compute the mean and standard deviation for below:

a) Sample data: 1,2 and 3
b) Population data: 10,20 and 30 

Answer:

a. Sample Data 1, 2, 3

i. Mean (x̄) = μ = (Σx) / n = (1 + 2 + 3) / 3 = 6 / 3 = 2

Mean = 2

ii. Standard deviation (sample data): S = √s^2 = √(Σ(x - x̄)^2) / (n - 1)

(1 - 2)² = 1

(2 - 2)² = 0

(3 - 2)² = 1

Σ(x - x̄)^2 = 1 + 0 + 1 = 2

n -1 = 2

S = √(Σ(x - x̄)^2) / (n - 1) = √2/2 = 1

Sample. Standard Deviation = 1

b. Population Data: 10, 20 and 30:

i. Mean (μ) = (Σx) / n = (10 + 20 + 30) / 3 = 60 / 3 = 20

Mean (μ) = 20

ii. Standard deviation (Population) for 10, 20, 30

σ = √(σ^2) = √(Σ(x - μ)^2) / N
(10 - 20)² = 100
(20 - 20)² = 0
(30 - 20)² = 100
(Σ(x - μ)^2) = 200
N = 3
σ = √200/3 = 8.1649 = 8.16

Population. Standard Deviation = 8.16 

Q2. A researcher conducted an experiment with a sample size of 36. The sample standard deviation was found to be 12. What is the standard error in this case? 

Answer:

Standard error of mean = $\frac{1}{\sqrt{n}} \times \text{standard deviation}$

Q3: There are two stocks X and Y. You have invested 50% of your wealth in each of these stocks. The covariance of the two stocks is 0.1. The variance of stock X is 0.1 and that of stock Y is 0.2. What is the correlation?

Correlation: r = Cov(X, Y) / (σX * σY)

Correlation = Covariance (X, Y) / (Standard Deviation of Stock X * Standard Deviation of Stock Y)

As per provided details:

Covariance = 0.1
Variance of Stock X (σX) = 0.1
Variance of Stock Y (σY) = 0.2

Standard Deviation (Stock X) = √0.1 = 0.316
Standard Deviation (Stock Y) = √0.2 = 0.447

Correlation = 0.1 / (0.316 * 0.447)
≈ 0.1 / 0.141352 = 0.707

Thus, correlation between stock X and stock Y = 0.707.

Q4: Find a symmetrically distributed interval around μ that will include 95% of the sample means when μ = 368, σ = 15, and n = 25.

Answer:

Confidence Interval = μ ± (Z * (σ / √n))

μ (population mean) = 368  

95% confidence level corresponds to a Z-score of approximately 1.96  

σ (population standard deviation) = 15  

n (Sample size) = 25

Standard error = (σ / √n) = 15 / √25 = 15 / 5 = 3  

⇒ Standard error = 3

Confidence Interval = 368 ± (1.96 * (15 / √25))  

= 368 ± (1.96 * 3) = 368 ± 5.88

Thus, the symmetrically distributed interval around μ = 368 ± 5.88 = (362.12, 373.88)

Q5: You are interested in the average response time of customer support calls at a telecommunications company. You randomly select a sample of 100 customer support calls and find that the mean response time is 8 minutes, with a standard deviation of 3 minutes. Calculate a 90% confidence interval for the average response time of customer support calls at the company.

Answer:

Confidence Interval = x̄ ± (Z * (s / √n))

x̄ (sample mean) = 8
Z is the Z-score corresponding to a 90% confidence is 1.645
s (sample standard deviation) = 3
n = 100

Confidence Interval = 8 ± (1.645 * (3 / √100))
Standard Error = 3 / √100 = 3 / 10 = 0.3
Margin of Error = 0.4935

⇒ Confidence Interval = 8 ± (1.645 * (3 / √100)) = 8 ± 0.4935

Confidence Interval = 8 ± 0.4935

Therefore, the average response time of customer support calls at the company should lie between 7.5065 and 8.4935 minutes. (@90% confidence)

Q6: Is the average number of sales per month for a new sales strategy significantly different from the average number of sales per month for the previous sales strategy? (Consider alpha = 0.05) 

Answer: Below is detail from excel for t-Test: Two-Sample Assuming Equal Variances

We can calculate the sample means and sample standard deviations for each group.

New Sales Strategy:

Mean 1 = (50 + 55 + 60 + 65 + 70) / 5 = 60

Standard Deviation (sd1) = 7.07

Previous Sales Strategy:

Mean 2 = (45 + 48 + 50 + 52 + 55) / 5 = 50

Standard Deviation (sd2) = 3.40

t ≈ (60 - 50) / 3.53 ≈ 2.83

Comparing t-value with the critical t-value at alpha 0.05 and (df) = (n1 + n2 - 2).

i.e.: df = 5 + 5 - 2 = 8.

Calculated critical t-value at alpha = 0.05 and df = 8 is 2.306.

Calculated t-value (2.83) is greater than the critical t-value (2.306), thus we reject the null hypothesis.

Therefore, we can observe to have that the average number of sales per month with new sales strategy

is significantly different from the average number of sales per month for the previous sales strategy at a

significance level of 0.05.

Q7. 7. A company wants to determine if the average salary of its employees has increased over the past

10 years, taking into account both full-time and part-time employees. They take a random sample of 50

employees, 25 full-time and 25 part-time, to see whether there is a significant change from the average

salary of $50,000 ten years ago. Perform Hypothesis testing (one sample t-test) to come to a

conclusion. (Consider alpha = 0.05)

Answer:

Hypothesis Setup

Null Hypothesis (H0): The average salary of the employees is equal to $50,000. (H0: μ = $50,000)

Alternative Hypothesis (Ha): The average salary of the employees is greater than $50,000. (Ha: μ > $50,000)

Test Statistics Calculation

Mean salary of the sample: $52,440

Sample variance: $591,108.57

Sample size (n): 50

Hypothesized mean (μ0): $50,000

Degrees of freedom (df): 49

t-Statistic: 0.7096

Step 3: P-Value Calculation

P(T <= t) one-tail = 0.2406

t Critical one-tail (at alpha 0.05) = 1.6766

==Calculation==

Data:

• Sample Mean ($\bar{x}$) = $52,440
• Hypothesized Mean ($\mu_0$) = $50,000
• Sample Variance ($s^2$) = 591,108,571.4
• Number of Observations (n) = 50
• Significance Level ($\alpha$) = 0.05

Formulating Hypotheses

• Null Hypothesis (H0): $\mu = 50,000$ (The average salary is $50,000)
• Alternative Hypothesis (Ha): $\mu > 50,000$ (The average salary is greater than $50,000)

Standard Error of the Mean (SE)

The standard error of the mean is calculated using the formula:

$$SE = \frac{s}{\sqrt{n}}$$

Where:

• $s$ is the standard deviation of the sample, which is the square root of the sample variance.
• $n$ is the number of observations.

First, calculate the sample standard deviation ($\mathrm{s):}$

$$s=\sqrt{591,108,571.4}\approx 24,307.59$$

Now, calculate the standard error:

$$SE = \frac{24,307.59}{\sqrt{50}} \approx \frac{24,307.59}{7.071} \approx 3,436.48$$

t-Statistic

The t-statistic is calculated using the formula:

$$t = \frac{\bar{x} - \mu_0}{SE}$$

Substitute the values:

$$t = \frac{52,440 - 50,000}{3,436.48} \approx \frac{2,440}{3,436.48} \approx 0.71$$

Step 4: Determine the Critical t-Value

Since this is a one-tailed test with $n - 1 = 50 - 1 = 49$ degrees of freedom and $\alpha = 0.05$, the critical t-value can be found using a t-distribution table or calculator. For $df = 49$ and $\alpha = 0.05$:

$$t_{\text{critical}} \approx 1.6766$$

Comparing t-Statistic to Critical t-Value

The calculated t-statistic (0.71) is less than the critical t-value (1.6766).

p-Value

Using the t-distribution, the p-value associated with the t-statistic of 0.71 is approximately 0.2406 (for one-tailed test).

====

Conclusion

Since the p-value (0.2406) is greater than the significance level (alpha = 0.05), we fail to reject the null hypothesis (H0).

There is no significant evidence to suggest that the average salary of the employees has increased from the $50,000 average 10 years ago. The data does not support the claim that the average salary is greater than $50,000.