Exam 9: Inferences Based on Two Samples

The degrees of freedom associated with the pooled t test, based on sample sizes 10 and 12 are

In an experiment designed to study the effects of illumination level on task performance, subjects were required to insert a fine-tipped probe into the eyeholes of ten needles in rapid succession both for a low light level with black background and a higher level with a white background. Each data value is the time (sec) required to complete the task. Subject Black 25.01 41.05 27.47 25.74 24.96 28.84 25.85 20.89 32.05 White 16.61 24.98 24.59 19.68 16.07 20.84 18.23 19.50 22.96 Does the data indicate that the higher level of illumination yields a decrease of more than 5 sec in true average task completion time? Test the appropriate hypotheses using the P-value approach.

When the necessary conditions are met in testing $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0 \text { versus } H _ {0} : p _ { 1 } - p _ { 2 } \neq 0$ the two sample proportions are $\hat { p } _ { 1 } = .40 \text { and } \hat { p } _ { 2 } = .30 \text {, and } V \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = .0016 \text { when } H _ { o }$ is true. Then, the value of the test statistic is

Two independent samples of sizes 15 and 17 are randomly selected from two normal populations with equal variances. Which of the following distributions should be used for developing confidence intervals and for testing hypotheses about the difference between the two population means $\left( \mu _ { 1 } - \mu _ { 2 } \right) ?$

At the .05 significance level, the null hypothesis $H _ { o } : \mu _ { D } \geq 0$ is rejected in a paired t test, where the data consists of 15 independent pairs, if

Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$ be a random sample from a population with mean $\mu _ { 1 } \text { and variance } \sigma _ { 1 } ^ { 2 } \text {, and let } Y _ { 1 } , Y _ { 2 } , \ldots \ldots , Y _ { n }$ be a random sample from a population with mean $\mu _ { 2 } \text { and variance } \sigma _ { 2 } ^ { 2 } \text {, }$ and that the X and Y samples are independent of one another. Which of the following statements are not true?

Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$ with X and Y independent variables, and let $\hat { p } _ { 1 } = X / m \text { and } \hat { p } _ { 2 } = Y / \mathrm { n } \text {. Then } \mathrm { E } \left( \hat { \mathrm { p } } _ { 1 } - \hat { \mathrm { p } } _ { 2 } \right) =\underline{\quad\quad}$ ,so $\hat { p } _ { 1 } - \hat { p } _ { 2 }$ is an __________ estimator of $p _ { 1 } - p _ { 2 }$

Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$ be a random sample from a population with mean $\mu _ { 1 } \text { and variance } \sigma _ { 1 } ^ { 2 } \text {, and let } Y _ { 1 } , Y _ { 2 } , \ldots \ldots , Y _ { n }$ be a random sample with mean $\mu _ { 2 } \text { and variance } \sigma _ { 2 } ^ { 2 } \text {, }$ and that the X and Y samples are independent of one another. The expected value of $\bar { X } - \bar { Y }$ is __________ and the standard deviation of $\bar { X } - \bar { Y } \text { is }σ _ { \bar { x } - \bar { y } }$ = __________.

Provided that the sample sizes m and n of two independent samples X and Y are both large , then a confidence interval for the difference between the two population means, $\mu _ { 1 } - \mu _ { 2 } ,$ with a confidence level of approximately $100 ( 1 - \alpha ) \%$ is __________, where the values of the population variances are unknown.

The number of degrees of freedom for a paired t test, where the data consists of 10 independent pairs, is equal to

A study comparing different types of batteries showed that the average lifetimes of Duracell Alkaline AA batteries and Eveready Energizer Alkaline AA batteries were given as 4.5 hours and 4.2 hours, respectively. Suppose these are the population average lifetimes. a. Let $\bar { X }$ be the sample average lifetime of 150 Duracell batteries and $\bar { Y }$ be the sample average lifetime of 150 Eveready batteries. What is the mean value of $\bar { X } - \bar { Y }$ (i.e., where is the distribution of $\bar { X } - \bar { Y }$ centered)? How does your answer depend on the specified sample sizes? b. Suppose the population standard deviations of lifetime are 1.8 hours for Duracell batteries and 2.0 hours for Eveready batteries. With the sample sizes given in part (a), what is the variance of the statistic $\bar { X } - \bar { Y }$ , and what is its standard deviation? c. For the sample sizes given in part (a), what is the approximate distribution curve of $\bar { X } - \bar { Y }$ (include a measurement scale on the horizontal axis)? Would the shape of the curve necessarily be the same for sample sizes of 10 batteries of each type? Explain.

Which of the following statements are not correct assumptions for developing pooled confidence intervals and for testing hypotheses about the difference between two population means $\left( \mu _ { 1 } - \mu _ { 2 } \right) ?$

The sample standard deviation of sodium concentration in whole blood (mEq/L) for m = 20 marine eels was found to be $s _ { 1 } = 40.5 ,$ whereas the sample standard deviation of concentration for n = 20 freshwater eels was $s _ { 2 } = 32.5$ . Assuming normality of the two concentration distributions, test at level .10 to see whether the data suggests any difference between concentration variances for the two types of eels.