Exam 22: Statistical Inference: Conclusion

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. $\text { Old-style tapes }$$\quad\quad\text { 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 difference in the proportions of unacceptable tapes between the old and new tapes.

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. $\text { Old-style tapes }$$\quad\quad\text { 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 Determine whether these data are sufficient to infer at the 10% significance level that the two population variances differ.

When the necessary conditions are met, a two-tail test is being conducted at $\alpha$ = 0.025 to test H₀: $\sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 }$ = 1. The two sample variances are $s _ { 1 } ^ { 2 }$ = 375 and $s _ { 2 } ^ { 2 }$ = 625, and the sample sizes are n₁ = 36 and n₂ = 36. The calculated value of the test statistic will be F = 0.60.

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

Two samples of sizes 22 and 18 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: A 39 B 40 C 38 D 41

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. Sample of semester 1 final marks 65 45 53 76 53 85 55 63 85 77 45 57 55 60 83 96 83 55 52 67 82 64 62 88 71 Sample of semester 2 final marks 45 40 53 58 75 46 82 54 75 59 45 54 87 77 63 81 60 56 53 65 52 65 60 65 54 Semester 1 Mean 65.48 Stan dard Error 2.679 Median 63 Mode 55 Standard Deviation 13.395 Sample Variance 179.43 Range 43 Minimum 45 Maximum 88 Sum 1637 Count 25 Semester 2 Mean 60.96 Standard Error 2.5136 Median 59 Mode 54 Standard Deviation 12.568 Sample Variance 157.96 Range 47 Minimum 40 Maximum 87 Sum 1524 Count 25 (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.

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

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.

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

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.

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

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

In constructing a 95% interval estimate for the ratio of two population variances, $\sigma _ { 1 } ^ { 2 }$ / $\sigma _ { 2 } ^ { 2 }$ , 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.

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. $\text { Old-style tapes }$$\quad\quad\text { 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 a. Do the data allow us to infer at the 10% significance level that the new-technology tapes are superior to the old-style tapes in terms of the number of unacceptable tapes? b. Find the p-value of the test in a. and explain how to use it to test the hypotheses.

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

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 Do these results allow us to conclude at the 5% significance level that the variance of the percentage improvement with approach 2 is less than 40?

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.

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.

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.

Two independent samples of sizes 50 and 50 are randomly selected from two populations to test the difference between the population means, $\mu _ { 1 } - \mu _ { 2 }$ . The sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is: A normally distributed. B approximately normal. C t -distributed with 98 degrees of freedom. D chi-squared distributed with 99 degrees of freedom.