Exam 22: Statistical Inference: Conclusion

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

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 Can we conclude at the 10% significance level that the mean number of re-recordings of the new tapes is at least 55?

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

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. Method 1 Method 2 =86 =98 =324 =841 Estimate with 95% confidence the mean bacteria count with method 2.

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.

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 mean number of re-recordings of the new tapes.

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.

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.

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?

In a hypothesis test for the population variance, the hypotheses are: $H _ { 0 } : \sigma ^ { 2 } = 175$ $H _ { 1 } : \sigma ^ { 2 } \neq 175$ If the sample size is 25 and the test is being carried out at the 5% level of significance, the rejection region will be:

In testing the null hypothesis $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0$ , if $\mathcal { H } _ { 0 }$ is false, the test could lead to:

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.

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.

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

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

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

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.

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 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 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: