STAT217 Worksheet #1

1) Find the following:

a) P(0 < Z < 1.58) (44.29%)

b) P(-2.8 < Z < 0) (49.74%)

c) P(-1.28 < Z < 0.67) (64.83%)

d) P(1.39 < Z < 2.54) (7.68%)

e) P(-1.98 < Z < -0.02) (46.81%)

f) The value of x such that P(Z < x) = 0.8508 (1.04)

g) The value of x such that P(-x < Z < x) = 0.9756 (2.25)

2) Suppose X is normally distributed with a mean of 5 and a standard deviation of 2. Find the following:

a) P(X < 7) (84.13%)

b) P(X > 9.2) (1.79%)

c) P(4 < X < 6.4) (44.95%)

d) The value of x such that P(X < x) = 0.937 (8.06)

e) Between which two values would 95.44% of the distribution lie? (1 and 9)

3) Suppose the running time for the 100m dash is normally distributed with a mean of 10.2 seconds and a standard deviation of 0.2 seconds.

a) What is the probability a runner’s time is under 10.01 seconds? (17.11%)

b) What is the probability a runner’s time is more than 10.3 seconds? (30.85%)

c) What is the cutoff time for a runner to place in the fastest 5%? (9.87 seconds)

4) Surveys indicate that adults spend an average of 4 hours per day between watching TV and listening to radio with a standard deviation of 1.2 hours. If a sample of 50 adults is chosen, what is the probability the average amount of time this group spends doing the above activities is more than 4.3 hours? (3.84%)

5) Suppose the sample size is 100 instead. What does the probability in question 4 change to? (0.62%)

6) A survey of 100 people was taken to see how much they spend on fast food per week. For the 100 people, the average was $15.14 with a standard deviation of $2.54.

a) Construct a 95% confidence interval of the average amount that people spend per week on fast food (14.64 < m < 15.64)

b) Suppose the level of confidence was increased to 99%. Does the confidence interval become narrower or wider? (14.49 < m < 15.79; wider)

c) Suppose the sample size was 200 instead of 100 and the sample mean and standard deviation remained at $15.14 and $2.54 respectively. Would a 95% confidence interval be narrower or wider than the one in part a? (14.79 < m < 15.49; narrower)

d) Suppose the sample size was 100 but the standard deviation was $5.08. If the sample mean remained at $15.14, would a 95% confidence interval be narrower or wider than the one in part a? (14.14 < m < 16.14; wider)

e) Suppose they wanted to conduct another study. If they use the standard deviation of $2.54 and they wanted the margin of error to be 50 cents and the level of confidence to be 95%, what would the sample size need to be? (n = 100)

7) In a gated community of 800 households, a survey was conducted to determine the average amount spent per year on home improvements. Of the 120 households that responded, the average was $2,650 with a standard deviation of $720. Construct a 95% confidence interval of the average amount spent per year on home improvements by households in this community. Round the limits to the nearest dollar. ($2,531 < µ < $2,769)

8) For a subsequent study, what would be the required sample size at 95% confidence if they use the standard deviation from the initial study and the margin of error is $50? (400)

9) A research firm does monthly surveys for the Workers Compensation Board and usually get about 180 cases a month. Suppose that in the past the percentage of people generally satisfied with the Board was 85%. What would be the necessary sample size if they want the margin of error to be no more than 2% at a 95% level of confidence? (158)

10) Suppose the monthly quota for this survey is 80 cases (i.e. n=80). What would be the margin of error at a 95% level of confidence? Assume p=0.85. (5.85%)

11) Suppose the Compensation Board doesn't want a margin of error greater than 5%. If p=0.85 and the quota is 80 cases, what would be the approximate level of confidence? (90.7%)

From the textbook

Section 6-2 Q29 (page 314) and Q32 (page 315)

Section 6-4 Q29 (page 337) and Q31 (page 338)