Exam 9: Inferences Based on a Two Samples: Confidence Intervals and Tests of Hypotheses

The sample standard deviation of differences sd is equal to the difference of the sample standard deviations $s _ { 1 } - s _ { 2 }$

The FDA is comparing the mean caffeine contents of two brands of cola. Independent random samples of 6 -oz. cans of each brand were selected and the caffeine content of each can determined. The study provided the following summary information. Brand A Brand B Sample size 15 10 Mean 18 20 Variance 1.2 1.5 How many cans of each soda would need to be sampled in order to estimate the difference in the mean caffeine content to within 5 with $95 \%$ reliability? A) $n _ { 1 } = n _ { 2 } = 42$ B) $n _ { 1 } = n _ { 2 } = 18$ C) $n _ { 1 } = n _ { 2 } = 57$ D) $n _ { 1 } = n _ { 2 } = 21$

A marketing study was conducted to compare the variation in the age of male and female purchasers of a certain product. Random and independent samples were selected for both male and female purchasers of the product. The sample data is shown here: Female: n = 31, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 21, sample mean = 39.80, sample standard deviation = 10.040 Calculate the test statistics that should be used to determine if the variation in the female ages exceeds the variation in the male ages.

A confidence interval for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ is $( 5,8 )$ . Which of the following inferences is correct? A) $\mu _ { 1 } > \mu _ { 2 }$ B) $\mu _ { 1 } < \mu _ { 2 }$ C) $\mu _ { 1 } = \mu _ { 2 }$ D) no significant difference between means

A confidence interval for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ is $( - 5,8 )$ . Which of the following inferences is correct? A) no significant difference between means B) $\mu _ { 1 } > \mu _ { 2 }$ C) $\mu _ { 1 } < \mu _ { 2 }$ D) $\mu _ { 1 } = \mu _ { 2 }$

Calculate the degrees of freedom associated with a small-sample test of hypothesis for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ , assuming $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ and $n _ { 1 } = n _ { 2 } = 12$

Given $v _ { 1 } = 30 \text { and } v _ { 2 } = 60 , \text { find } P ( F < 1.68 )$

Consider the following set of salary data: Men (1) Women (2) Sample Size 100 80 Mean \ 12,850 \ 13,000 Standard Deviation \ 345 \ 500 What assumptions are necessary to perform a test for the difference in population means?

Which of the following represents the difference in two population means?

A paired difference experiment yielded $\mathrm { n } _ { \mathrm { d } }$ pairs of observations. For the given case, what is the rejection region for testing $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 9$ against $\mathrm { Ha } : \mu _ { \mathrm { d } } \neq 9$ ? $\mathrm { n } _ { \mathrm { d } } = 15 , \alpha = 0.10$ A) $| t | > 1.761$ B) $| t | > 1.753$ C) $\mathrm { t } > 1.345$ D) $\mathrm { t } > 1.761$

A researcher is investigating which of two newly developed automobile engine oils is better at prolonging the life of an engine. Since there are a variety of automobile engines, 20 different engine types were randomly selected and were tested using each of the two engine oils. The number of hours of continuous use before engine breakdown was recorded for each engine oil. Based on the information provided, what type of analysis will yield the most useful information?

University administrators are trying to decide where to build a new parking garage on campus. The state legislature has budgeted just enough money for one parking structure on campus. The administrators have determined that the parking garage will be built either by the college of engineering or by the college of business. To help make the final decision, the university has randomly and independently asked students from each of the two colleges to estimate how long they usually take to find a parking spot on campus (in minutes). Suppose that the sample sizes selected by the university for the two samples were both $\mathrm { n } _ { \mathrm { e } } = \mathrm { n } _ { \mathrm { b } } = 15$ . What critical value should be used by the university in the calculations for the 95% confidence interval for ?e - ?b? Assume that the university used the pooled estimate of the population variances in the calculation of the confidence interval.

A paired difference experiment has 15 pairs of observations. What is the rejection region for testing Ha: μd > 0? Use α = .05.

Construct a 90% confidence interval for $\left( p _ { 1 } - p _ { 2 } \right) \text { when } n _ { 1 } = 400 , n _ { 2 } = 550 , \hat { p } _ { 1 } = .42 , \text { and } \hat { p } _ { 2 } = .63$

The owners of an industrial plant want to determine which of two types of fuel (gas or electricity) will produce more useful energy at a lower cost. The cost is measured by plant investment per delivered quad ($ invested /quadrillion BTUs). The smaller this number, the less the industrial plant pays for delivered energy. Suppose we wish to determine if there is a difference in the average investment/quad between using electricity and using gas. Our null and alternative hypotheses would be: A) $H _ { 0 } : \left( \mu _ { e } - \mu _ { g } \right) = 0$ vs. $H _ { a } : \left( \mu _ { e } - \mu _ { g } \right) \neq 0$ B) $H _ { 0 } : \left( \mu _ { e } - \mu _ { g } \right) = 0$ vs. $H _ { a } : \left( \mu _ { e } - \mu _ { g } \right) > 0$ C) $H _ { 0 } : \left( \mu _ { e } - \mu _ { g } \right) = 0$ vs. $H _ { a } : \left( \mu _ { e } - \mu _ { g } \right) < 0$ D) $H _ { 0 } : \left( \mu _ { e } - \mu _ { g } \right) = 0$ vs. $H _ { a } : \left( \mu _ { e } - \mu _ { g } \right) = 0$

In an exit poll, 42 of 75 men sampled supported a ballot initiative to raise the local sales tax to build a new football stadium. In the same poll, 41 of 85 women sampled supported the initiative. Find and interpret the p-value for the test of hypothesis that the proportions of men and women who support the initiative are different.

A researcher is investigating which of two newly developed automobile engine oils is better at prolonging the life of an engine. Since there are a variety of automobile engines, 20 different engine types were randomly selected and were tested using each of the two engine oils. The number of hours of continuous use before engine breakdown was recorded for each engine oil. Suppose the following 95% confidence interval for μA - μB was calculated: (100, 2500). Which of the following inferences is correct?

A paired difference experiment yielded $\mathrm { n } _ { \mathrm { d } }$ pairs of observations. For the given case, what is the rejection region for testing $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 9$ against Ha: $\mu _ { \mathrm { d } } > 9$ ? $\mathrm { n } _ { \mathrm { d } } = 23 , \alpha = 0.05$ A) $\mathrm { t } > 1.717$ B) $\mathrm { t } > 1.714$ C) $\mathrm { t } < 1.717$ D) t $< 2.074$

In order to compare the means of two populations, independent random samples of 144 observations are selected from each population with the following results. Sample 1 Sample 2 =7,123 =6,957 =175 =225 Use a 95% confidence interval to estimate the difference between the population means (μ1 - μ2). Interpret the confidence interval.

A marketing study was conducted to compare the mean age of male and female purchasers of a certain product. Random and independent samples were selected for both male and female purchasers of the product. It was desired to test to determine if the mean age of all female purchasers exceeds the mean age of all male purchasers. The sample data is shown here: Female: n = 20, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 20, sample mean = 39.80, sample standard deviation = 10.040 Use the pooled estimate of the population standard deviation to calculate the value of the test statistic to use in this test of hypothesis.