Exam 22: Statistical Inference: Conclusion

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:

When the necessary conditions are met, a two-tail test is being conducted at $\alpha$ = 0.10 to test H₀: $\sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 }$ = 1. The two sample variances are $s _ { 1 } ^ { 2 }$ = 736 and $s _ { 2 } ^ { 2 }$ = 1024, and the sample sizes are n₁ = 16 and n₂ = 25. The rejection region is F > 2.11 or F < 0.4367.

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.

There are different approaches to fitness training. To judge which one of two approaches is better, 200 25-year-old men were randomly selected to participate in an experiment. For four weeks, 100 men were trained according to approach 1 while the other 100 men were trained according to approach 2. The percentage improvement in fitness was measured for each man and the statistics shown below were computed. The percentage figures are known to be normally distributed. Approach 1 Approach 2 =27.3 =33.6 =47.614 =28.09 Estimate with 95% confidence the variance of the percentage improvement with approach 1.

The equal-variances test statistic of $\mu _ { 1 } - \mu _ { 2 }$ is Student t-distributed with n₁ + n₂ - 2 degrees of freedom, provided that the two sample sizes are equal.

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

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 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.

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

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.

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.

The pooled-variance estimator, $s _ { p } ^ { 2 }$ , requires that the two population variances be equal.

Both the equal-variances and unequal-variances t-test statistics of $\mu _ { 1 } - \mu _ { 2 }$ require that the two populations be Student t-distributed.

There are different approaches to fitness training. To judge which one of two approaches is better, 200 25-year-old men were randomly selected to participate in an experiment. For four weeks, 100 men were trained according to approach 1 while the other 100 men were trained according to approach 2. The percentage improvement in fitness was measured for each man and the statistics shown below were computed. The percentage figures are known to be normally distributed. Approach 1 Approach 2 =27.3 =33.6 =47.614 =28.09 a. Estimate with 95% confidence the mean percentage improvement with approach 2. b. Do these results allow us to conclude at the 5% significance level that the mean percentage improvement with approach 1 is at least 25%?

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:

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 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:

There are different approaches to fitness training. To judge which one of two approaches is better, 200 25-year-old men were randomly selected to participate in an experiment. For four weeks, 100 men were trained according to approach 1 while the other 100 men were trained according to approach 2. The percentage improvement in fitness was measured for each man and the statistics shown below were computed. The percentage figures are known to be normally distributed. Approach 1 Approach 2 =27.3 =33.6 =47.614 =28.09 a. Estimate with 95% confidence the ratio of the variances of the percentage improvement in fitness, and briefly describe what the interval estimate tells you. b. Do these results allow us to conclude at the 5% significance level that approach 2 is superior?

Videocassette recorder (VCR) tapes are designed so that users can repeatedly record new material over old material. However, after a number of re-recordings the tape begins to deteriorate. A VCR tape manufacturer is experimenting with a new technology, which hopefully will produce longer-lasting tapes. Thirty of the old-style tapes and 30 utilising the new technology were used in an experiment. The tapes were used to record and re-record programs until they began to deteriorate. The number of re-recordings is assumed to be normally distributed. It is generally accepted that the number of re-recordings should exceed 55. Any tapes that do not meet this criterion are considered to be unacceptable. The number of re-recordings were observed and shown in the accompanying table. Old-style tapes New-technology tapes 60 61 48 68 70 58 51 46 66 74 72 69 66 63 61 77 73 49 73 55 71 59 66 61 71 49 76 52 58 59 47 56 55 66 51 49 60 62 64 62 59 57 52 51 63 51 56 66 64 68 52 50 55 76 47 55 58 63 68 78 Estimate with 90% confidence the population variance of the number of re-recordings of the new tape.

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?