Exam 9: Inferences Based on Two Samples

Let $X _ { 1 } , \ldots \ldots , X _ { 8 }$ be a random sample from a normal distribution with variance $\sigma _ { 1 } ^ { 2 } \text {, and let } Y _ { 1 } , \ldots \ldots , Y _ { 10 }$ be another random sample (independent of the $\left. X _ { 2 } ^ { \prime } s \right)$ from a normal distribution with variance $\sigma _ { 1 } ^ { 2 } \text {, and let } S _ { 1 } ^ { 2 } \text { and } S _ { 2 } ^ { 2 }$ denote the two sample variances. Then the random variable $F = \left( S _ { 1 } ^ { 2 } / \sigma _ { 1 } ^ { 2 } \right) \left( S _ { 1 } ^ { 2 } / \sigma _ { 1 } ^ { 2 } \right)$ has an F distribution with $v _ { 1 } =$$\underline{\quad\quad}$ and $v _ { 2 } =$$\underline{\quad\quad}$

Investigators are often interested in comparing the effects of two different treatments on a response. If the individuals or subjects to be used in the comparison are not assigned by the investigators to the two treatments, the study is said to be __________. If the investigators assign individuals or subjects to the two treatments in a random fashion, this is referred to as __________.

For an F distribution with parameters $v _ { 1 } \text { and } v _ { 2 } , \text { where } v _ { 1 }$ is the number of numerator degrees of freedom, and $v _ { 2 }$ is the number of denominator degrees of freedom, which of the following statements are true?

Ionizing radiation is being given increasing attention as a method for preserving horticultural products. A study reports that 153 of 180 irradiated garlic bulbs were marketable (no external sprouting, rotting, or softening) 240 days after treatment, whereas only 117 of 180 untreated bulbs were marketable after this length of time. Does this data suggest that ionizing radiation is beneficial as far as marketability is concerned?

Which of the following statements are true?

The degrees of freedom associated with the pooled t test, based on sample sizes m and n, is given by __________.

A summary data on proportional stress limits for specimens constructed using two different types of wood are shown below: Type of wood Sample size Sample mean Sample St. Dev. Red oak 14 8.50 80 Douglas fir 10 6.65 1.28 Assuming that both samples were selected from normal distributions, carry out a test of hypotheses to decide whether the true average proportional stress limit for red oak joints exceeds that for Douglas fir joints by more than one Mpa?

In testing $H _ { o } : \mu _ { 1 } - \mu _ { 2 } = 0 \text { versus } H _ { a } : \mu _ { 1 } - \mu _ { 2 } \neq 0 \text {, }$ the computed value of the test statistic is z = 2.25. The P-value for this two-tailed test is then __________.

Analogous to the notation $f _ { \kappa , v } \text { and } \chi _ { \kappa , \nu } ^ { 2 } \text {, we use } F _ { \kappa , v , \nu }$ for the point on the axis that captures __________ of the area under the F density curve with $v _ { 1 } \text { and } v _ { 2 }$ degrees of freedom in the __________ tail.

Consider the accompanying data on breaking load (kg/25 mm width) for various fabrics in both an unabraded condition and an abraded condition. Use the paired t test at significance level .01 to test $H _ { 0 } : \mu _ { D } = 0 \text { versus } H _ { a } : \mu _ { D } > 0$ . Fabric U 25.6 48.8 49.8 43.2 38.7 55.0 36.4 51.5 26.5 52.5 46.5 36.5 34.5 20.0 28.5 46.0

To decide whether two different types of steel have the same true average fracture toughness values, $n$ specimens of each type are tested, yielding the following results: Type Sample Average Sample St. Dev. 1 60.2 1.0 2 60.0 1.0 Calculate the $P$ -value for the appropriate two-sample $z$ test, assuming that the data was based on $n$ = 100. Then repeat the calculation for $n$ = 400. Is the small p-value for $n$ = 400 indicative of a difference that has practical significance? Would you have been satisfied with just a report of the p-value? Comment briefly.

In calculating 95% confidence interval for $\mu _ { 1 } - \mu _ { 2 } ;$ the difference between the means of two normally distributed populations, summary statistics from two independent samples are: $m = 10 , \bar { x } = 50 , s _ { 1 } ^ { 2 } = .64 , n = 10 , \bar { y } = 40 \text {, and } s _ { 2 } ^ { 2 } = 1.86$ Then, the upper limit of the confidence interval is

A random sample of 5726 telephone numbers from a certain region taken in March 2002 yielded 1105 that were unlisted, and 1 year later a sample of 5384 yielded 980 unlisted numbers. a. Test at level .10 to see whether there is a difference in true proportions of unlisted numbers between the two years. b. If $p _ { 1 } = .20 \text { and } p _ { 2 } = .18$ what sample sizes (m = n) would be necessary to detect such a difference with probability .90?

Which of the following statements are true?

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

In a study of copper deficiency in cattle, the copper values (ug Cu/100mL blood) were determined both for cattle grazing in an area known to have well-defined molybdenum anomalies (metal values in excess of the normal range of regional variation) and for cattle grazing in a nonanomalous area, resulting in $s _ { 1 } = 21.5$ (m = 48) for the anomalous condition and $s 2 = 19.45$ (n = 45) for the nonanomalous condition. Test for the equality versus inequality of population variances at significance level .10 by using the P-value approach.

The rejection region for level .025 paired t test in testing $H _ { o } : \mu _ { D } \leq 0 \text { versus } H _ { \mathrm { d } } : \mu _ { D } > 0$ is __________, where the data consists of 12 independent pairs.

A study includes the accompanying data on compression strength (lb) for a sample of 12-oz aluminum cans filled with strawberry drink and another sample filled with cola. Does the data suggest that the extra carbonation of cola results in a higher average compression strength? Base your answer on a $P$ -value. What assumptions are necessary for your analysis? Beverage Sample Size Sample Mean Sample St. Dev. Strawberry drink 15 546 21 Cola 15 560 15

The pooled t confidence interval for estimating $\mu _ { 1 } - \mu _ { 2 }$ with confidence level $100 ( 1 - \alpha ) \%$ using two independent samples X and Y with sizes m and n is given by __________.

Let $X _ { 1 } , \ldots \ldots , X _ { 20 }$ be a random sample from a normal distribution with variance $\sigma _ { 1 } ^ { 2 } , \mathrm { let } Y _ { 1 } , \ldots \ldots , Y _ { 25 }$ be another random sample (independent of the $\left. X _ { 2 } ^ { \prime } s \right)$ from a normal distribution with variance $\sigma _ { 1 } ^ { 2 } , \text { and let } S _ { 1 } ^ { 2 } \text { and } S _ { 1 } ^ { 2 }$ denote the two sample variances. Which of the following statements are not true in testing $H _ { 0 } : σ _ { 1 } ^ { 2 } = σ _ { 2 } ^ { 2 } ,$ where the test statistic value is $f = s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ and the test is performed at .10 level?