Sampling Distribution of sample mean,X ¯ and sample proportion, p^{^}

1. GPA is normally distributed in the population with μ = 4.2 and σ = 0.6, and a random sample of 17 students is selected.

I. Find the probability that the average GPA is

a) at least 4.5

b) between 4.0 and 4.5

II. Find the average GPA of the top 10% of students

2. In the Australian Federal election on 24 Nov 2007, 53% of Australians voted for the Australian Labor Party (ALP). Assuming that voting intentions have not changed, if we now choose a random sample of 80 people, what is the probability that

I. more than 45% intend to vote for the ALP?

II. between 38% and 45% intend to vote for the ALP?

3. Lifts usually have signs indicating their maximum capacity. Consider a sign in a lift that reads "maximum capacity 1120kg or 16 persons". Suppose that the weights of lift-users are normally distributed with a mean of 68kg and a standard deviation of 8kg.

(a) What is the probability that a lift-user will weigh more than 70kg?

(b) What is the probability that a lift-user will weigh between 65 and 75kg?

(c) What is the probability that 16 people will exceed the weight limit of 1120kg?

(d) What is the probability 25 that people will not exceed the weight limit? Clearly explain how you calculate (or estimate) this probability.

[Hint for parts (c) and (d): Convert the probability statement about total weight into a probability statement about average weight.]

4. For a random variable that is normally distributed, with mean of 80 and standard deviation of 10, determine the probability that a simple random sample of 25 items will have a mean that is

a. greater than 78.

b. between 79 and 85.

c. less than 85.

5. In a special Morgan Gallup Poll on employment conducted in a certain suburb on November 12/13, 1997, a cross-section of 631 men and women aged 14 or over were interviewed by telephone. Forty-five percent of those interviewed expected unemployment to increase. Construct a 95% interval estimate of the proportion of the entire population that expect unemployment to increase. Interpret this interval.

Confidence Interval to Estimate μ Using the z Statistic (σ known)

6. The amount of time for students to complete an assignment in class has a standard deviation of 5 minutes. From a sample of 35 students, the average time to complete the assignment was 55 minutes.

a) Find a 90% confidence interval for the true mean time to complete the assignment. Interpret your answer.

b) Does the population amount of time have to be normally distributed here? Explain.

Not necessarily. Based on the Central Limit Theorem, even if the population time is not normal, the sampling distribution of the mean will be normal because n>30

Confidence Interval to Estimate μ Using the t Statistic (σ unknown)

7. The average capacity usage for iPhone users has been estimated as 400 megabytes per month. Assuming this finding to be based on a simple random sample of 80 iPhone users, with a sample standard deviation of 90 megabytes per month, construct and interpret 95% and 99% confidence intervals for the population mean usage per month. Given these confidence intervals, would it seem very unusual if another sample of this size were to have a mean of 350.0 megabytes per month?