Module 2

Question 1

Central Limit Theorem states that…?

Answer 1

if we take enough sufficiently large samples from any population, the means of those samples will be normally distributed, regardless of the shape of the underlying population.

Question 2

For a standard normal distribution (µ=0, σ=1), the area under the curve less than 1.25 is 0.894.

What is the approximate percentage of the area under the curve less than -1.25?
A - 0.894
B - 0.394
C - 0.211
D - 0.106

Answer 2

D - 0.106
1–0.894=0.106 is the area under the curve for all values greater than 1.25. Since the normal distribution is symmetric, 0.106 is also the area under the curve for all values less than -1.25.

Question 3

According to the Central Limit Theorem, the means of random samples from which of the following distributions will be normally distributed, assuming the samples are sufficiently large?

A- The heights of basketball players
B - the sum of 2 dice
C - The annual income of HBS Online graduates
D - All of the above

Answer 3

D - All of the above
According to the Central Limit Theorem, if we take large enough samples, the distribution of sample means will be normally distributed regardless of the shape of the underlying population.

Question 4

If the mean weight of all students in a class is 165 pounds with a variance of 234.09 square pounds, what is the z-value associated with a student whose weight is 140 pounds?

A - 1.63
B - 0.11
C - -0.11
D - -1.63

Answer 4

D - -1.63

Question 5

A journalist wants to determine the average annual salary of CEOs in the S&P 1,500. He does not have time to survey all 1,500 CEOs but wants to be 95% confident that his estimate is within $50,000 of the true mean. The journalist takes a preliminary sample and estimates that the standard deviation is approximately $449,300.

What is the minimum number of CEOs that the journalist must survey to be within $50,000 of the true average annual salary? Remember that the z-value associated with a 95% confidence interval is 1.96. (Please enter your answer as an integer; that is, as a whole number with no decimal point.)

Answer 5

311

Question 6

You report a confidence interval to your boss but she says that she wants a narrower range. 
SELECT ALL of the ways you can reduce the width of the confidence interval.
A - Increase the sample size
B - Decrease the sample size
C - Increase the confidence level
D - Decrease the confidence level
E - Increase the mean
F - Decrease the mean

Answer 6

A - Increase the sample size

D - Decrease the confidence level

Question 7

Which of the following is the MOST LIKELY result of using a survey with biased questions?

A - The standard deviation of the sample will be larger than the standard deviation of the population.
B - The standard deviation of the sample will be smaller than the standard deviation of the population.
C - The data in your sample will differ in a systematic way from data based on unbiased random selections from the population.
D - The data in your sample will not follow a normal distribution.

Answer 7

C - The data in your sample will differ in a systematic way from data based on unbiased random selections from the population.

In general, surveys with biased questions may lead to biased data, which differ systematically from what would be seen in an unbiased sample. For example, biased survey questions would lead to systematic differences between the answers given on your surveys and the answers that would be that would be given on a more neutral survey.

Question 8

A z-value of a point x is what?

Answer 8

The distance x lies from the mean, measured in standard deviations

Question 9

Is the following question biased or unbiased?

“Should cyclists be required to wear helmets?”

Answer 9

Unbiased.

Question 10

Is the following question biased or unbiased?

How much would you be willing to pay for a new car?

Answer 10

Unbiased

Question 11

The probability of all values less than or equal to a particular value is called what?

Answer 11

A cumulative probability

Question 12

If the average IQ is 100 and the standard deviation is 15, approximately what percentage of people have IQs above 130?

Answer 12

2.5%

130 is two standard deviations above the mean (130-100=30=215=2stdev). We know that approximately 95% of the distribution is within 2 standard deviations of the mean. Therefore 5% must fall beyond 2 standard deviations, 2.5% at the top and 2.5% at the bottom.

Question 13

If a particular standardized test has a mean score of 500 and standard deviation of 100, what percentage of test-takers score between 500 and 600?

Answer 13

34%

100 is one standard deviation above the mean (600-500 =100= 1100 = 1stdev). We know that approximately 68% of the distribution is within 1 standard deviation of the mean. Therefore 34% must fall beyond 1 standard deviation above the mean.

Question 14

What formula would you use to answer the following question?

If the mean of a normally distributed population is -10 with a standard deviation of 2, what is the likelihood of obtaining a value less than or equal to -7?

Answer 14

=NORM.DIST

=.93

Question 15

A store owner is interested in opening a second shop. She wants to estimate the true average daily revenue of her current shop to decide whether expanding her business is a good idea. The store owner takes a random sample of 60 days over a six-month period and finds that the mean revenue of those days is 3,472.00 dollars with variance 315,900.20 square dollars. Calculate a 95% confidence interval to estimate the true average daily revenue.315

Answer 15

Lower: $3,329.78
Upper: $3,614.22

Question 16

A curious student in a large economics course is interested in calculating the percentage of his classmates who scored lower than he did on the GMAT; he scored 490. He knows that GMAT scores are normally distributed and that the average score is approximately 540. He also knows that 95% of his classmates scored between 400 and 680. Based on this information, calculate the percentage of his classmates who scored lower than he did.

Mean 540
2.5% Bound 400
97.5% Bound 680
Student’s Score 490

Answer 16

Since GMAT scores are normally distributed, we know that P(μ–1.96σ ≤ x ≤ μ+1.96σ) = 95%. Thus, to find the standard deviation, subtract the lower bound from the mean and divide by 1.96. The standard deviation of the distribution is (B1-B2)/1.96 = (540-400)/1.96 = 71.4. (Note that because the normal curve is symmetrical, we could calculate the same value using (B3-B1)/1.96 = (680-540)/1.96 = 71.4). To find the cumulative probability, P(x ≤ 490), use the Excel function NORM.DIST(x, mean, standard_dev, TRUE). Here, NORM.DIST(B4,B1,71.4,TRUE) = NORM.DIST(490,540,71.4,TRUE) = 0.24, or 24%. Approximately 24% of his classmates scored lower than he did. You must link directly to the values in order to obtain the correct answer

Question 17

What formula would you use to input a dummy variable for the options “Yes” or “No”, where 1 indicates yes and 0 indicates no.

Answer 17

=IF(A2=”Yes”,1,0)