Question 1

A manufacturing plant has two assembly lines for producing glass bottles. The plant manager was concerned about whether the proportion of defective bottles differs between the two lines. Two independent random samples were selected and the following summary data computed:  Find a 95% confidence interval for the true difference in proportion of defective bottles produced by the two assembly lines.
______________
Based on the interval above, can one conclude there is a difference in proportion of defective bottles produced by the two lines?
______________
Explain.
________________________________________________________

Accepted Answer

The answer of A manufacturing plant has two assembly lines...

Question 2

If you wish to estimate the difference between two population means when the population variances are known, the 90% confidence interval estimate can be developed using which of the following critical values of z?
A) 2.33
B) 1.28
C) 1.96
D) 2.58
E) 1.645

Accepted Answer

For a 90% confidence interval, the critical z-value is 1.645, which corresponds to the middle 90% of the standard normal distribution.

Question 3

In developing a confidence interval estimate for the difference between two population means, an increase in the size of the samples will result in:
A) a wider confidence interval
B) a narrower confidence interval
C) a smaller critical value z
D) the same confidence interval
E) none of these

Accepted Answer

Increasing the sample size decreases the standard error, which leads to a narrower confidence interval.

Question 4

Increasing the confidence level for a confidence interval estimate for the difference between two population means, with all other things held constant, will result in a wider confidence interval estimate.

Accepted Answer

When the confidence level increases, the margin of error also increases, which in turn widens the confidence interval estimate.

Question 5

In estimating the difference between two population means, if a 90% confidence interval estimate includes zero, then we can conclude that there is a 90% chance that the difference between the two population means is zero.

Accepted Answer

The statement is incorrect. A confidence interval estimate that includes zero means that there is not enough evidence to support a difference between the two population means at the given level of confidence. It does not mean that there is a chance of zero difference.

Question 6

The difference between two sample means  is an unbiased estimator of the difference between two population means  .

Accepted Answer

The difference between two sample means is an unbiased estimator of the difference between two population means, as long as the samples are randomly selected and independent, and the populations have equal variances. This is a fundamental concept in statistical inference.

Question 7

When two independent random samples of sizes  and  have been selected from populations with means  and  , and variances  and  , which of the following is true?
A) The mean ofis.
B) The standard error ofis.
C) The standard error ofis estimated as.
D) All of these.
E) None of these.

Accepted Answer

The mean of the difference between two independent samples can be calculated as:

µ1-µ2

The standard error of the difference between two independent samples can be estimated as:

√((s1^2/n1) + (s2^2/n2))

Therefore, both A and B are true. 
C is also true as it represents the formula for estimating the standard error. 
Thus, the correct choice is D, i.e., all of these are true.

Question 8

Suppose a 90% confidence interval for the mean time it takes to serve a customer at a drive-in bank is 120 seconds to 220 seconds. At the 90% confidence level, there is not enough evidence to conclude that the mean service time is not 200 seconds.

Accepted Answer

Since the confidence interval contains the value of 200 seconds, we cannot reject the null hypothesis that the mean service time is 200 seconds. Therefore, there is not enough evidence to conclude that the mean service time is not 200 seconds at the 90% confidence level.

Question 9

The best estimator of the difference between two population means  is the difference between two sample means  .

Accepted Answer

The formula for the difference between two population means involves unknown population parameters, whereas the formula for the difference between two sample means involves known sample statistics. Therefore, the latter is a better estimator.

Question 10

Instead of paying to support welfare recipients, many Californians want them to find jobs; if necessary, they want the state to create public service jobs for those who cannot find jobs in private industry. In a survey of 800 registered voters, 400 Republicans and 400 Democrats, 75% of the Republicans and 90% of the Democrats favored the creation of public service jobs. Use a large-sample estimation procedure to compare the proportions of Republicans and Democrats who favor creating public service jobs in the population of registered voters in California.
The approximate 95% confidence interval is:
______________
Based on the interval above, is there a difference in the proportion of Republicans and Democrats who favor creating public service jobs in California?
______________
Explain.
________________________________________________________

Accepted Answer

The answer of Instead of paying to support welfare recipients,...

Question 11

If you wish to estimate the difference between two population means when the population variances are known, the 99% confidence interval estimate can be developed using which of the following critical values of z?
A) 2.575
B) 1.645
C) 2.325
D) 1.275
E) 2.582

Accepted Answer

The critical value of z for a 99% confidence level with two-tailed test is 2.575.

Question 12

Suppose a 95% confidence interval for the mean height of a 12-year-old male in the United States is 54 to 65 inches. In repeated sampling, 95% of the intervals constructed will contain the interval from 54 to 65 inches.

Accepted Answer

In repeated sampling, 95% of the confidence intervals constructed will contain the true mean height of a 12-year-old male in the United States, not necessarily the interval from 54 to 65 inches.

Question 13

Two samples are said to be independent if they are selected at different points in time.

Accepted Answer

Two samples are said to be independent if the selection of one sample does not influence or affect the selection of the other sample. The timing of their selection is not a defining factor for independence.

Question 14

Suppose a 90% confidence interval for the mean time it takes to serve a customer at a drive-in bank is 120 seconds to 220 seconds. In repeated sampling 90% of the intervals constructed using the appropriate formula will contain the actual mean time.

Accepted Answer

This statement accurately describes the interpretation of a 90% confidence interval. In repeated sampling, if we were to construct 100 intervals, we would expect 90 of them to contain the true population mean.

Question 15

A stylist at The Hair Care Palace gathered data on the number of hair colorings given on Saturdays and on weekdays. Her results are listed below. Assume the two samples were independently taken from normal populations. Find the point estimate of p₁ - p₂. ______________ Find the margin of error. ______________ Estimate the difference in the true proportions with a 99% confidence interval. ______________ Interpret this interval. ________________________________________________________

Accepted Answer

The answer of A stylist at The Hair Care Palace...

Question 16

In a study of the relationship between birth order and college success, an investigator found that 140 in a sample of 200 college graduates were firstborn or only children. In a sample of 120 non-graduates of comparable age and socioeconomic background, the number of firstborn or only children was 66. Estimate the difference between the proportions of firstborn or only children in the two populations from which these samples were drawn. Use a 90% confidence interval and interpret your results.
The approximate 90% confidence interval is:
______________
Based on the interval above, is there a difference between the proportions of firstborn or only children in the two populations?
______________
Explain.
________________________________________________________

Accepted Answer

The answer of In a study of the relationship between...

Question 17

The standard error of the sampling distribution of  is given by the formula:
SE =  .

Accepted Answer

The formula for the standard error of a sampling distribution depends on various factors such as the sample size, population standard deviation, and type of sampling. Therefore, it cannot be determined by just knowing the name or code of a variable.

Question 18

When two independent random samples of sizes  and  have been selected from populations with means  and  , and variances  and  , which of the following is true?
A) If the sampled populations are normally distributed, then the sampling distribution ofis exactly normal only whenandare both 30 or more.
B) If the sampled populations are normally distributed, then the sampling distribution ofis exactly normal regardless of the sizes ofand.
C) If the sampled populations are not normally distributed, then the sampling distribution ofis approximately normally distributed regardless of the sizes ofand.
D) If the sampled populations are not normally distributed, then the sampling distribution ofis approximately normally distributed only ifis 30 or more.
E) None of these.

Accepted Answer

The answer of When two independent random samples of sizes...

Question 19

In constructing a 95% 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:  ,  ,  ,  ,  , and  . The upper confidence limit is:
A) 18.78
B) 6.78
C) 5.76
D) 77.3
E) 89.3

Accepted Answer

Since the sample sizes are small and the population variances are assumed to be unequal, we need to use the t-distribution. 
The formula for the confidence interval is:
$$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2,df} \cdot \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$$
where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $s_1$ and $s_2$ are the sample standard deviations, $n_1$ and $n_2$ are the sample sizes, $df$ is the degrees of freedom, and $t_{\alpha/2,df}$ is the critical value from the t-distribution with $df$ degrees of freedom and a significance level of $\alpha/2$. 
Plugging in the given values, we get: 
$$\bar{x}_1 - \bar{x}_2 = 8.74$$
$$s_1 = 0.52, s_2 = 3.84$$
$$n_1 = n_2 = 5$$
To find the degrees of freedom, we use the following formula: 
$$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}$$
Plugging in the values, we get $df = 6.69$. 
Using a t-table or calculator, we find the critical value to be $t_{\alpha/2,df} = 2.447$. 
Plugging all the values into the formula, we get: 
$$(8.74) \pm 2.447 \cdot \sqrt{\frac{0.52^2}{5}+\frac{3.84^2}{5}}$$
Simplifying, we get: 
$$(8.74) \pm 6.78$$
Therefore, the upper confidence limit is $8.74 + 6.78 = 15.52$, which is closest to choice B) 6.78.

Question 20

When two independent random samples of sizes  and  have been selected from populations with means  and  and variances  and  , the standard error of the sampling distribution of  the difference between the two sample means, is found by taking the square root of the sum of the two population variances; namely,  .

Accepted Answer

The standard error of the sampling distribution of the difference between two sample means is found by taking the square root of the sum of the two population variances divided by their respective sample sizes (i.e. using the pooled variance formula).

Quiz 8: Large-Sample Estimation

If you wish to estimate the difference between two population means when the population variances are known, the 90% confidence interval estimate can be developed using which of the following critical values of z?

In developing a confidence interval estimate for the difference between two population means, an increase in the size of the samples will result in:

Increasing the confidence level for a confidence interval estimate for the difference between two population means, with all other things held constant, will result in a wider confidence interval estimate.

In estimating the difference between two population means, if a 90% confidence interval estimate includes zero, then we can conclude that there is a 90% chance that the difference between the two population means is zero.

The difference between two sample means is an unbiased estimator of the difference between two population means .

When two independent random samples of sizes and have been selected from populations with means and , and variances and , which of the following is true?

Suppose a 90% confidence interval for the mean time it takes to serve a customer at a drive-in bank is 120 seconds to 220 seconds. At the 90% confidence level, there is not enough evidence to conclude that the mean service time is not 200 seconds.

The best estimator of the difference between two population means is the difference between two sample means .

If you wish to estimate the difference between two population means when the population variances are known, the 99% confidence interval estimate can be developed using which of the following critical values of z?

Suppose a 95% confidence interval for the mean height of a 12-year-old male in the United States is 54 to 65 inches. In repeated sampling, 95% of the intervals constructed will contain the interval from 54 to 65 inches.

Two samples are said to be independent if they are selected at different points in time.

Suppose a 90% confidence interval for the mean time it takes to serve a customer at a drive-in bank is 120 seconds to 220 seconds. In repeated sampling 90% of the intervals constructed using the appropriate formula will contain the actual mean time.

The standard error of the sampling distribution of is given by the formula: SE = .

When two independent random samples of sizes and have been selected from populations with means and , and variances and , which of the following is true?