Exam 11: Estimation: Comparing Two Populations

In constructing a 99% confidence interval estimate for the difference between the means of two normally distributed populations, where the unknown population variances are assumed not to be equal, summary statistics computed from two independent samples are as follows: n1 = 28 $\bar { X } _ { 1 }$ = 123 S1 = 8.5 N2 = 45 $\bar { X } _ { 2 }$ = 105 S2 = 12.4 The lower confidence limit is:

Two random samples from two normal populations produced the following statistics: n1 = 51, $\bar { X } _ { 1 }$ = 35, s1 = 28. n2 = 40, $\bar { X } _ { 2 }$ = 28, s2 = 10. Assume that the two population variances are different. Estimate with 95% confidence the difference between the two population means.

We cannot estimate the difference between population means by estimating the mean difference μD, when the data are produced by a matched pairs experiment.

A study is trying to estimate the difference between the annual salaries paid to female and male employees working for the same large company. They take a random sample of 50 females and find that their average a nnual salary is $75 600 with a variance of $21 250. They take a random sample of 50 males and find that their average annual salary is $78 500 with a variance of $22 500. Find and interpret a 95% confidence interval for the difference in annual salaries for female and male employees of this large company.

In constructing a confidence interval estimate for the difference between the means of two normally distributed populations, using two independent samples, we:

The following data were generated from a matched pairs experiment: Pair: 1 2 3 4 5 6 7 Sample 1: 8 15 7 9 10 13 11 Sample 2: 12 18 8 9 12 11 10 Estimate with 90% confidence the mean difference.

We find the following statistics: n1 = 400, x1 = 105. n2 = 500, x2 = 140. Estimate with 90% confidence the difference between the two population proportions.

When the two population variances are unequal, we cannot pool the data and produce a common estimator. We must calculate s12 or s22 and use them to estimate σ12 and σ22 respectively.

Two independent random samples of 25 observations each are drawn from two normal populations. The parameters of these populations are: Population 1: \mu=150,\sigma=50 Population 2: \mu=130,\sigma=45 Find the probability that the mean of sample 1 will exceed the mean of sample 2.

If two random samples, each of size 36, are selected independently from two populations with variances of 42 and 50, then the standard error of the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , equals 2.5556.

In constructing a confidence interval estimate for the difference between two population proportions, we:

A simple random sample of ten firms was asked how much money (in thousands of dollars) they spent on employee training programs this year and how much they plan to spend on these programs next year. The data are shown below. Firm 1 2 3 4 5 6 7 8 9 10 This year 25 31 12 15 21 36 18 5 9 17 Next year 21 30 18 20 22 36 20 10 8 15 Assume that the populations of amount spent on employee training programs are normally distributed. a. Estimate with 95% confidence the mean difference. b. Briefly explain what the interval estimate in part a. tells you.

A political poll taken immediately prior to a state election reveals that 158 out of 250 male voters and 105 out of 200 female voters intend to vote for the Independent candidate. Estimate with 95% confidence the difference between the proportions of male and female voters who intend to vote for the Independent candidate.

In order to draw inferences about p₁ − p₂, we take two independent samples − a sample of size n1 from population 1 and a sample of size n2 from population 2.

In random samples of 25 and 22 from each of two normal populations, we find the following statistics: $\bar { X } _ { 1 }$ = 56, s1 = 8. $\bar { X } _ { 2 }$ = 62, s2 = 8.5. Assume that the population variances are equal. Estimate with 95% confidence the difference between the two population means.

Which of the following best describes a matched pairs experiment?

The marketing manager of a pharmaceutical company believes that more females than males use its acne medicine. In a recent survey, 2500 teenagers were asked whether or not they use that particular product. The responses, categorised by gender, are summarised below. Use acne medicine Don't use acne medicine Female 540 810 Male 391 759 a. Estimate with 90% confidence the difference in the proportions of male and female users of the acne medicine. b. Describe what the interval estimate in part a. tells you.

Suppose that the starting salaries of finance graduates from university A are normally distributed with a mean of $36 750 and a standard deviation of $5320. The starting salaries of finance graduates from university B are normally distributed with a mean of $34 625 and a standard deviation of $6540. If simple random samples of 50 finance graduates are selected from each university, what is the probability that the sample mean of university A graduates will exceed that of university B graduates?

For a matched pairs experiment, find a 90% confidence interval for µD given that sample mean differences is 5, the standard deviation of differences is 3 and the sample sizes are 30.

Two samples are selected at random from two independent normally distributed populations. Sample 1 has 49 observations and has a mean of 10 and a standard deviation of 5. Sample 2 has 36 observations and has a mean of 12 and a standard deviation of 3. The standard error of the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , is: