# MATHEMATICS

- During routine conversations, the CEO of a new start up reports that 15% of adults between the ages of 21 and 39 will purchase her new product. Hearing this, some investors decide to conduct a large scale study, hoping to estimate the proportion to within 5% with 90% confidence. How many randomly selected adults between the ages of 21 and 39 must they survey?

The number of adults that should be surveyed is 138 (Round up to the nearest whole number).

- A random sample of 20 purchases showed the amounts in the table (in $). The mean is $52.08 and the standard deviation is $20.92.

a) What is the standard error of the mean?

b) How would the standard error change if the sample size had been 5 instead of 20? (Assume that the sample standard deviation didn’t change).

Table:

80.70 45.42 46.89 83.80

28.19 48.65 29.17 50.75

31.17 72.20 32.29 52.41

72.43 33.13 74.04 87.73

16.41 38.40 58.79 59.00

a) The standard error of the mean is 4.68(Round to two decimal places as needed)

b) How would the standard error change if the sample size was 5 instead of 20 with the same sample standard deviation? Select the correct choice below and fill in any answer boxes within your choice.

a) The standard error would increase. The new standard error would be 2 times the old.

b) The standard error would decrease. The new standard error would be the old standard error divided by _____________

c) The standard error would not change.

- Top management at a large Software company wishes to estimate the average number of hours its firm’s professional employees volunteer in the local community. Based on past similar studies, the standard deviation was found to be 2.22 hours. If top management wants to estimate the average number of hours volunteered per month by their professional staff to within one hour with 99% confidence, how many randomly selected professional employees would they need to sample?

a) 19

b) 44

c) 33

d) 25

e) 54

- A young investor believes that he can beat the market by pricing stocks that will increase in value. Assume that on average 52% of the stocks selected by a portfolio manager will increase over 12 months. Of the 29 stocks that the young investor bought over the last 12 months, 17 have increased. Can he claim that he is better at predicting increases that the typical portfolio manager? Test at a = 0.05

What is the null and alternative hypothesis for this test?

a)

b)

c)

d)

Calculate the test statistic.

Z = 0.714(Round to three decimal places as needed)

What is the P-value for the test statistic?

P-Value =0.238(Round to three decimal places).

What can the investor conclude? Assume a=0.05

a) He rejects the null hypothesis and cannot claim that he is better at predicting increases than the typical portfolio manager.

b) He fails to reject the null hypothesis and can claim that he is better at predicting increases than the typical portfolio manager, but this conclusion may not be reliable. ‘

c) He fails to reject the null hypothesis and can not claim that he is better at predicting increases than the typical portfolio manager, but this conclusion may not be reliable.

d) He rejects the null hypothesis and can claim that he is better at predicting increases than the typical portfolio manager, but this conclusion may not be reliable.

- A random sample of the amounts for 16 purchases was taken. The mean was $44.37, the standard deviation was $23.32, and the margin of error for a 95% confidence interval was $12.43.

a) To reduce the margin of error to about $6, how large would the sample size have to be?

b) How large would the sample size have to be to reduce the margin of error to $1.2?

a) The new sample size should be69(Round up to the nearest integer)

b) The new sample size should be 1717(Round up to the nearest integer).