Deck 22: Statistical Inference: Conclusion

Question 1

For a sample of size 25 observations taken from a normally distributed population. The sample standard deviation is 6, a 95% confidence interval estimate for the population mean would require the use of:

A) z = 1.96
B) t = 1.711
C) z = 1.645
D) t = 2.064

Answer 1

D

Question 2

A random sample of 20 observations taken from a normally distributed population revealed a sample mean of 65 and a sample variance of 16. The lower limit of a 90% confidence interval for the population mean would equal:

A) 66.546.
B) 63.454.
C) 63.812.
D) 66.188.

Answer 2

B

Question 3

Two independent samples of sizes 50 and 50 are randomly selected from two populations to test the difference between the population means, . The sampling distribution of the sample mean difference is:

A) normally distributed.
B) approximately normal.
C) t-distributed with 98 degrees of freedom.
D) chi-squared distributed with 99 degrees of freedom.

Answer 3

B

Question 4

A sample of size 300 had 96 successes. The lower limit of the 99% confidence interval for the population proportion is:

A) 0.3728.
B) 0.2672.
C) 0.2506.
D) 0.3894.

Question 5

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. The two sample proportions are $\hat { p } _ { 1 }$ = 0.21 and $\hat { p } _ { 2 }$ = 0.15, and the standard error of the sampling distribution of are $\hat { p } _ { 1 }$ - $\hat { p } _ { 2 }$ is 0.018. The calculated value of the test statistic will be:

A) t = -3.33.
B) z = 3.33.
C) t = 3.33.
D) None of these choices are correct.

Question 6

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions, but your statistical software provides only a one-tail area of 0.03 as part of its output. The p-value for this test will be:

A) 0.03
B) 0.06
C) 0.05
D) None of these choices are correct.

Question 7

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions, testing at the 5% level of significance. Which of the following is the p-value for this test if the calculated z test statistic is 1.34?

A) 0.0250
B) 0.1802
C) 0.0901
D) 0.9099

Question 8

A random sample of size 15 taken from a normally distributed population resulted in a sample variance of 25. The upper limit of a 99% confidence interval for the population variance would be:

A) 12.868.
B) 92.032.
C) 85.896.
D) 75.100.

Question 9

Assuming that all necessary conditions are met, what needs to be changed in the formula so that we can use it to construct a confidence interval estimate for the difference of two population means when the population variances are assumed to be equal?

A) The should be replaced by .
B) The z should be replaced by .
C) The z should be replaced by .
D) The should be replaced by .

Question 10

In testing the null hypothesis , if is false, the test could lead to:

A) Type I error.
B) Type II error.
C) Either a Type I or a Type II error.
D) None of these choices are correct.

Question 11

In a hypothesis test for the population variance, the hypotheses are: . . If the sample size is 15 and the test is being carried out at the 5% level of significance, the null hypothesis will be rejected if:

A) .
B) .
C) .
D) .

Question 12

In constructing a 95% interval estimate for the ratio of two population variances, / , two independent samples of sizes 30 and 40 are drawn from the populations. If the sample variances are 425 and 675, then the upper confidence limit is about:

A) 1.2215.
B) 0.3132.
C) 1.2656.
D) 0.3246.

Question 13

In testing for the equality of two population variances, when the populations are normally distributed, the 5% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of:

A) 0.950.
B) 0.050.
C) 0.025.
D) 0.100.

Question 14

In testing the difference between two population means for which the population variances are unknown and assumed to be equal, two independent samples are drawn from the populations. Which of the following tests is appropriate?

A) z-test.
B) Equal-variances t-test.
C) F-test.
D) Matched pairs t-test.

Question 15

In constructing a 90% 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: , , . , , . The lower confidence limit is:

A) 30.086.
B) 8.542.
C) 0.914.
D) 31.458.

Question 16

Two independent samples of sizes 20 and 25 are randomly selected from two normal populations with equal variances. In order to test the difference between the population means, the test statistic is:

A) a standard normal random variable.
B) approximately standard normal random variable.
C) Student t distributed with 45 degrees of freedom.
D) Student t distributed with 43 degrees of freedom.

Question 17

In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known and the calculated test statistic equals 1.05. If the test is upper-tail and the 10% level of significance has been specified, the conclusion should be to:

A) reject the null hypothesis.
B) not to reject the null hypothesis.
C) choose two other independent samples.
D) None of these choices are correct.

Question 18

Which of the following statements is correct regarding the percentile points of the F-distribution?

A) .
B) .
C) .
D) .

Question 19

A sample of size 200 from population 1 has 50 successes. A sample of size 200 from population 2 has 40 successes. The value of the test statistic for testing the null hypothesis that the proportion of successes in population 1 exceeds the proportion of successes in population 2 by 0.025 is:

A) 1.96.
B) 1.25.
C) 0.5998.
D) 1.20.

Question 20

Based on sample data, the 95% confidence interval limits for the population mean are LCL = 124.6 and UCL = 148.2. If the 5% level of significance were used in testing the hypotheses:
H₀ : $\mu$ = 150
H₁ : $\mu$$\neq$ 150,
The null hypothesis:

A) would not be rejected.
B) would be rejected.
C) would have to be revised.
D) There is insufficient information to decide whether to maintain or reject the null hypothesis.

Question 21

The pooled-variance estimator, , requires that the two population variances be equal.

Question 22

A one-tail test of the population proportion produces a test statistic z = -2.12. The p-value of the test is 0.034.

Question 23

In testing the hypotheses: H₀ : $\mu$ = 140
H₁ : $\mu$ $\neq$ 140,
Suppose that we rejected the null hypothesis at $\alpha$ = 0.10. Then for which of the following $\alpha$ values do we also reject the null hypothesis?

A) 0.025
B) 0.01
C) 0.05
D) All of these choices are correct.

Question 24

A sample of size 125 selected from one population has 55 successes, and a sample of size 140 selected from a second population has 70 successes. The test statistic for testing the equality of the population proportions is equal to:

A) -0.060.
B) -0.977.
C) -0.940.
D) -0.472.

Question 25

A random sample of 30 observations is selected from a normally distributed population. The sample variance is 12. In the 90% confidence interval for the population variance, the upper limit will be:

A) 15.176.
B) 8.177.
C) 19.652.
D) 16.941.

Question 26

Which of the following is a required condition for using the normal approximation to the binomial in constructing interval estimate for the difference between two population proportions?

A) and .
B) and .
C) and .
D) and .

Question 27

When the necessary conditions are met, a one-tail test is being conducted to test the difference between two population proportions, but your statistical software provides only a two-tail area of 0.058 as part of its output. The p-value for this test will be:

A) 0.029.
B) 0.971.
C) 0.029 or 0.972, depending on whether the test is a left-tail or a right-tail test.
D) 0.058.

Question 28

In a hypothesis test for the population variance, the hypotheses are: If the sample size is 25 and the test is being carried out at the 5% level of significance, the rejection region will be:

A) < 15.6587 or > 33.1963.
B) <12.4011 or >39.3641.
C) < 16.4734 or >34.3816.
D) < 13.1197 or <37.6525.

Question 29

Two samples of size 30 each are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is 59.

Question 30

Which of the following best describes a p-value?

A) A p-value is the probability of getting our population results or more extreme if the null hypothesis about the sample statistic were really true
B) A p-value is the probability of getting our sample results or more extreme if the null hypothesis about the population parameter were really true
C) A p-value is the probability of getting our sample results.
D) A p-value is the probability of getting our sample results or more extreme if the null hypothesis about the population parameter were really false

Question 31

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. The two sample proportions are = 0.32 and = 0.38, and the standard error of the sampling distribution of is 0.046. The calculated value of the test statistic will be 1.3043.

Question 32

Which of the following is the number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 30 pairs?

A) 30
B) 29
C) 28
D) 59

Question 33

When comparing two population variances, we test H₀: = 0.

Question 34

In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: , , . , , . Assume that the population variances are unequal. The standard error of the sampling distribution of the sample mean difference is equal to:

A) 0.3247.
B) 0.3282.
C) 0.1054.
D) 0.1125.

Question 35

From a sample of 500 items, 30 were found to be defective. The point estimate of the population proportion defective will be:

A) 0.06.
B) 30.0.
C) 16.667.
D) None of these choices are correct.

Question 36

If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10 level.

Question 37

In testing the hypotheses: , at the 10% significance level, if the sample proportion is 0.56, and the standard error of the sample proportion is 0.025, the appropriate conclusion is:

A) to reject .
B) not to reject .
C) to reject .
D) to reject both and .

Question 38

For a sample of 25 observations taken from a normally distributed population with standard deviation of 6, a 95% confidence interval estimate for the population mean would require the use of:

A) t = 2.064.
B) t = 1.711.
C) X² = 39.3641.
D) z = 1.96.

Question 39

Suppose that a one-tail t-test is being applied to find out if the population mean is at least 80. The level of significance is 0.10 and 25 observations were sampled. The rejection region is:

A) t > 2.492
B) t > 1.711
C) t < − 2.492
D) t > 1.318

Question 40

Two independent samples of sizes 35 and 40 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. In order to test the difference between the population means, , the sampling distribution of the sample mean difference, , is:

A) normally distributed.
B) t-distributed with 75 degrees of freedom.
C) t-distributed with 73 degrees of freedom.
D) F-distributed with 34 and 39 degrees of freedom.

Question 41

If a sample has 25 observations and a 99% confidence estimate for is needed, the appropriate t-score is 2.797.

Question 42

If a sample has 12 observations and a 90% confidence estimate for µ is needed, the appropriate t-critical value from the t tables is 1.796.

Question 43

If a sample of size 25 is selected, the value of A for the probability P(t_df=n-1 A) = 0.05 is 1.708.

Question 44

When the necessary conditions are met, a two-tail test is being conducted at = 0.025 to test
H₀: = 1. The two sample variances are = 375 and = 625, and the sample sizes are
n₁ = 36 and n₂ = 36. The calculated value of the test statistic will be F = 0.60.

Question 45

We use the F-test to determine whether two population variances are equal.

Question 46

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.0327 as part of its output. The p-value for this test will be 0.0654.

Question 47

The equal-variances test statistic of is Student t-distributed with n₁ + n₂ - 2 degrees of freedom, provided that the two sample sizes are equal.

Question 48

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. If the value of the test statistic z is 1.53, then the p-value is 0.126.

Question 49

The number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 8 pairs, is 14.

Question 50

If a null hypothesis about the population proportion p is rejected at the 0.05 level of significance, it must be rejected at the 0.01 level.

Question 51

The upper limit of the 89.9% confidence interval for the population proportion p, given that n = 80 and $\hat{P}$ = 0.40, is 0.4898.

Question 52

The lower limit of the 87.4% confidence interval for the population proportion p, given that n = 250 and $\hat{P}$ = 0.15, is 0.1492.

Question 53

If a sample of size 300 is selected, the value of A for the probability P(-A t_df=n-1 A) = 0.90 is 1.96.

Question 54

When the necessary conditions are met, a two-tail test is being conducted at = 0.10 to test H₀: = 1. The two sample variances are = 736 and = 1024, and the sample sizes are n₁ = 16 and n₂ = 25. The rejection region is F > 2.11 or F < 0.4367.

Question 55

If a sample has 300 observations and a 97.5% confidence estimate for p is needed, the appropriate z-score is 2.24.

Question 56

Descriptive statistics helps us describe and summarise data whereas inferential statistics helps us draw conclusions about populations based on samples of data.

Question 57

If a sample of size 28 is selected, the value of A for the probability P(-A t_df=n-1 t A) = 0.99 is 2.771.

Question 58

Both the equal-variances and unequal-variances t-test statistics of require that the two populations be Student t-distributed.

Question 59

If a sample has 20 observations and a 95% confidence estimate for is needed, the appropriate t-score is 1.729.

Question 60

In a one-tail test, the p-value is found to be equal to 0.0456. If the test had been two-tailed, the p-value would have been 0.0912.

Question 61

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Estimate a 95% confidence interval for the difference in the proportions of students who received a high distinction in semester 1 to semester 2.

Question 62

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. (a) Can we conclude at the 5% level of significance that semester 1 students have a higher proportion of high distinctions than semester 2 students, where a high distinction is a final mark greater than or equal to 85%?
(b) Find the p-value of the test, and explain how to use it to test the hypotheses.

Question 63

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Estimate and interpret a 95% confidence interval for the population average final mark for semester 1 students.

Question 64

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. (a) Determine whether these data are sufficient to infer at the 10% level of significance that the two population variances differ.
(b) Explain the decision of your test in part (a) in the context of this question.

Question 65

The irradiation of food to destroy bacteria is an increasingly common practice. In order to determine which one of two methods of irradiation is best, a scientist took a random sample of 100 one-kilogram packages of minced meat and subjected 50 of them to irradiation method 1 and the remaining 50 to irradiation method 2. The bacteria counts were measured and the following statistics were computed. The scientist noted that the data were normally distributed. Estimate with 95% confidence the difference in the mean bacteria counts between method 1 and method 2.

Question 66

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Can we conclude at the 5% significance level that the variance of semester 2 student's final marks is greater than 150?

Question 67

The sampling distributions we use for nominal (categorical) data are the Standard Normal distribution and the Chi-squared distribution.

Question 68

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Estimate and interpret a 95% confidence interval for the population average final mark for semester 2 students.

Question 69

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. There is a rumor going around the university that students with a higher IQ are enrolled in the semester 1 statistics course because they tend to be students enrolled in the degree with the higher entrance score for university. Can it be concluded at the 5% significance level that semester 1 students have a higher average final mark than semester 2 students? Assume that the population variances are unknown and equal.

Question 70

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Estimate and interpret a 95% confidence interval for the proportion of semester 2 students that passed the course.

Question 71

The irradiation of food to destroy bacteria is an increasingly common practice. In order to determine which one of two methods of irradiation is best, a scientist took a random sample of 100 one-kilogram packages of minced meat and subjected 50 of them to irradiation method 1 and the remaining 50 to irradiation method 2. The bacteria counts were measured and the following statistics were computed. The scientist noted that the data were normally distributed. Estimate with 95% confidence the mean bacteria count with method 2.

Question 72

The sampling distributions we use for numerical data are the Standard Normal distribution, the Student's t-distribution and the F distribution.

Question 73

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Can we conclude at the 5% significance level that the variance of semester 1 student's final marks is greater than 150?

Question 74

The irradiation of food to destroy bacteria is an increasingly common practice. In order to determine which one of two methods of irradiation is best, a scientist took a random sample of 100 one-kilogram packages of minced meat and subjected 50 of them to irradiation method 1 and the remaining 50 to irradiation method 2. The bacteria counts were measured and the following statistics were computed. The scientist noted that the data were normally distributed. $$\begin{array} { | c | c | } \hline \text { Method 1 } & \text { Method 2 } \\\hline \bar { x } _ { 1 } = 86 & \bar { x } _ { 2 } = 98 \\\hline s _ { 1 } ^ { 2 } = 324 & s _ { 2 } ^ { 2 } = 841 \\\hline\end{array}$$ Determine whether these data are sufficient to infer at the 5% significance level that the two population variances differ.

Question 75

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Estimate a 95% confidence interval for the difference in final marks between semester 1
and semester 2 students in this statistics course. Assume that the population variances are unknown
and equal.

Question 76

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. (a) Can we conclude at the 5% level of significance that over 40% of students in the population scored a pass grade in semester 1, where a pass grade is 50% to 64%?
(b) Find the p-value of the test and briefly explain how to use it to test the hypotheses.

Question 77

The sampling distribution of the random variable of interest is the source of statistical inference.

Question 78

With hypothesis testing, there are only two types of errors: Type I error where we incorrectly reject Ho and Type II error where we incorrectly retain Ho.

Question 79

The irradiation of food to destroy bacteria is an increasingly common practice. In order to determine which one of two methods of irradiation is best, a scientist took a random sample of 100 one-kilogram packages of minced meat and subjected 50 of them to irradiation method 1 and the remaining 50 to irradiation method 2. The bacteria counts were measured and the following statistics were computed. The scientist noted that the data were normally distributed. $$\begin{array} { | c | c | } \hline \text { Method 1 } & \text { Method 2 } \\\hline \bar { x } _ { 1 } = 86 & \bar { x } _ { 2 } = 98 \\\hline s _ { 1 } ^ { 2 } = 324 & s _ { 2 } ^ { 2 } = 841 \\\hline\end{array}$$ a. Estimate with 95% confidence the ratio of the variances of the bacteria counts under the two methods, and briefly describe what the interval estimate tells you.
b. Do these results allow us to infer at the 5% significance level that there is a difference in bacteria count between methods 1 and 2?
c. Do these results allow us to infer at the 5% significance level that the mean bacteria count with method 1 is less than 95?

Question 80

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. Estimate and interpret a 90% confidence interval of the ratio of population variances of final student marks from semester 1 and semester 2.