Exam 12: Estimation: Comparing Two Populations

The expected value of the difference of two sample means equals the difference of the corresponding population means: A only if the populations are normally distributed. B only if the samples are independent. C only if the populations are approximately normal and the sample sizes are large. D All of these choices are correct.

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?

Suppose that the starting salaries of male workers are normally distributed with a mean of $56 000 and a standard deviation of $12 000. The starting salaries of female workers are normally distributed with a mean of $50 000 and a standard deviation of $10 000. A random sample of 50 male workers and a random sample of 40 female workers are selected. a. What is the sampling distribution of the sample mean difference $\bar { X } _ { 1 }$ - $\bar { X } _ { 2 }$ ? Explain. b. Find the expected value and the standard error of the sample mean difference. c. What is the probability that the sample mean salary of female workers will not exceed that of the male workers?

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.

If two populations are not known to be normally distributed, the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , will be: A normally distributed only if both population sizes are greater than 30 . B normally distributed. C normally distributed only if at least one of the sample sizes is greater than 30 . D approximately normally distributed.

Estimate p₁ − p₂ with 99% confidence, given that n₁ = 50 and n₂ = 50 and the first sample has a proportion of 0.50 and the second sample has a proportion of 0.20

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: A. 0.1853 B. 0.7602 C. 0.7331 D. 0.8719

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: =28 x-=123 =8.5 =45 x-=105 =12.4 The lower confidence limit is: A. 24.485 B. 11.515. C. 13.116. D. 22.884

If two random samples of sizes $n _ { 1 }$ and $n _ { 2 }$ are selected independently from two populations with means $\mu _ { 1 }$ and $\mu _ { 2 }$ , then the mean of the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , equals: A. +. B. - C. / D. :

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

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.

If two random samples of sizes $n _ { 1 }$ and $n _ { 2 }$ are selected independently from two non-normally distributed populations, then the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , is: \begin{array}{|l|l|}\hline A&\text {always non-normal. }\\\hline B&\text {always normal. }\\\hline C&\text {approximately normal only if n_{1} and \( n_{2} \) are both larger than 30 . }\\\hline D&\text {approximately normal regardless of \( n_{1} \) and \( n_{2} \). }\\\hline \end{array}

In constructing a confidence interval estimate for the difference between the means of two normally distributed populations, using two independent samples, we: A pool the sample variances when the unknown population variances are equal. B pool the sample variances when the populati on variances are known and equal. C pool the sample variances when the population means are equal. D never pool the sample variances.

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.

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.

Which of the following statements is correct when estimating the difference between two population proportions p₁ − p₂? A We must take two independent samples. B We count the number of successes in each sample. C The sampling distribution of the difference of sample proportions is approximately normally distributed for large sample sizes. D All of these choices are correct.

If two random samples of sizes 30 and 45 are selected independently from two non-normal populations with means of 53 and 57, then the mean of the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , equals -4.

Two independent random samples are drawn from two normal populations. The sample sizes are 20 and 25, respectively. The parameters of these populations are: Population 1: \mu=505,\sigma=10. Population 2: \mu=4750,\sigma=7. Find the probability that the difference between the two sample means (X₁-bar - X₂-bar) is between 25 and 35.

Which of the following best describes a matched pairs experiment?

In constructing a confidence interval estimate for the difference between two population proportions, we: A pool the population proportions when the populations are normally distributed. B pool the population proportions when the population means are equal. C pool the population proportions when they are equal D never pool the population proportions.