Exam 22: Statistical Inference: Conclusion

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: $n _ { 1 } = 40$ , $\bar { x } _ { 1 } = 95$ , $s _ { 1 } = 12.5$ . $n _ { 2 } = 30$ , $\bar { x } _ { 2 } = 75$ , $s _ { 2 } = 35.5$ . The lower confidence limit is:

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.

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 Can we conclude at the 5% significance level that the variance of the bacteria count with method 2 is less than 1500?

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.

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 ratio of the variances of the number of re-recordings of the two types of tape, and briefly describe what the interval estimate tells you.

In a hypothesis test for the population variance, the hypotheses are: $H _ { 0 } : \sigma ^ { 2 } = 25$ . $H _ { 1 } : \sigma ^ { 2 } > 25$ . 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:

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

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

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:

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 Determine whether these data are sufficient to infer at the 5% significance level that the two population variances differ.

Which of the following best describes a p-value?

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 a. Do the data allow us to infer at the 10% significance level that the proportion of unacceptable new tapes is less than 20%? b. Can we infer at the 10% significance level that the variance of the number of re-recordings of the new tape is less than 100?

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 new tapes last longer than the old 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. 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 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.

In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: $n _ { 1 } = 20$ , $\bar { x } _ { 1 } = 10.8$ , $s _ { 1 } = 0.90$ . $n _ { 2 } = 18$ , $\bar { x } _ { 2 } = 9.6$ , $s _ { 2 } = 1.10$ . Assume that the population variances are unequal. The standard error of the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is equal to:

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

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 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?

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:

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.

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 ratio of the two variances.