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

Independent random samples, each containing 500 observations were selected from two binomial populations. The samples from populations 1 and 2 produced 210 and 320 successes, respectively. Test $H _ { 0 } : \left( p _ { 1 } - p _ { 2 } \right) = 0$ against $H _ { \mathrm { a } } : \left( p _ { 1 } - p _ { 2 } \right) < 0$ . Use $\alpha = .05$ .

The data for a random sample of five paired observations are shown below. Pair Observation 1 Observation 2 1 3 5 2 4 4 3 3 4 4 2 5 5 5 6 a. Calculate the difference between each pair of observations by subtracting observation 2 from observation 1 . Use the differences to calculate $\bar { d }$ and $s _ { d }$ . b. Calculate the means $\bar { x } _ { 1 }$ and $\bar { x } _ { 2 }$ of each column of observations. Show that $\bar { d } = \bar { x } _ { 1 } - \bar { x } _ { 2 }$ . c. Form a $90 \%$ confidence interval for $\mu _ { D }$ .

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 Identify the rejection region to that should be used to determine if the variation in the female ages exceeds the variation in the male ages when testing at α = 0.05.

$\text { The screens below show the results of a test of } H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) = 0 \text { against } H _ { a } : \left( \mu _ { 1 } - \mu _ { 2 } \right) \neq 0$ Comment on the validity of the results.

Consider the following set of salary data: Men (1) Women (2) Sample Size 100 80 Mean \ 12,850 \ 13,000 Standard Deviation \ 345 \ 500 Suppose the test statistic turned out to be z = -1.20 (not the correct value). Find a two-tailed p-value for this test statistic.

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: $\mathrm { n } = 10 , \quad$ sample mean $= 50.30 , \quad$ sample standard deviation $= 13.215$ Male: $\quad \mathrm { n } = 10 , \quad$ sample mean $= 39.80 , \quad$ sample standard deviation $= 10.040$ Which of the following assumptions must be true in order for the pooled test of hypothesis to be valid? I. Both the male and female populations of ages must possess approximately normal probability distributions. II. Both the male and female populations of ages must possess population variances that are equal. III. Both samples of ages must have been randomly and independently selected from their respective populations.

A consumer protection agency is comparing the work of two electrical contractors. The agency plans to inspect residences in which each of these contractors has done the wiring in order to estimate the difference in the proportions of residences that are electrically deficient. Suppose the proportions of residences with deficient work are expected to be about .6 for both contractors. How many homes should be sampled in order to estimate the difference in proportions using a 95% confidence interval of width .8? A) $n _ { 1 } = n _ { 2 } = 12$ B) $n _ { 1 } = n _ { 2 } = 29$ C) $n _ { 1 } = n _ { 2 } = 24$ D) $n _ { 1 } = n _ { 2 } = 6$

Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } = \sigma ^ { 2 } \text {. Calculate the pooled estimator of } \sigma ^ { 2 } \text { for } s _ { 1 } ^ { 2 } = 50 , s _ { 2 } ^ { 2 } = 57 \text {, and } n _ { 1 } = n _ { 2 } = 18 \text {. }$

Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } = \sigma ^ { 2 }$ . Calculate the pooled estimator of $\sigma ^ { 2 }$ for $s _ { 1 } ^ { 2 } = .88$ , $s _ { 2 } ^ { 2 } = 1.01$ , $n _ { 1 } = 10$ , and $n _ { 2 } = 12$

Calculate the degrees of freedom associated with a small-sample test of hypothesis for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ , assuming $\sigma _ { 1 } ^ { 2 } \approx \sigma _ { 2 } ^ { 2 }$ and $n _ { 1 } = n _ { 2 } = 20$

Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low tech industry with CEOs in the consumer products industry. HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM RETURN industry 1 (low tech) $\quad ($ NUMBER $= 15$ ) industry $3 \quad$ (consumer products) $\quad$ (NUMBER $= 15 )$ X= industry1 Y= industry 3 SAMPLE MEAN OF X =157.286 SAMPLE VARIANCE OF X=1563.45 SAMPLE SIZE OF X=14 SAMPLE MEAN OF Y=217.583 SAMPLE VARIANCE OF Y=1601.54 SAMPLE SIZE OF Y=12 MEAN X - MEAN Y =-60.2976 t=-4.23468 P-VALUE =0.000290753 P-VALUE /2=0.000145377 SD. ERROR =14.239 If we conclude that the mean return-to-pay ratios of the consumer products and low tech CEOs are equal when, in fact, a difference really does exist between the means, we would be making a __________.

Find F.10 where $v _ { 1 } = 20 \text { and } v _ { 2 } = 40$ .

A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests by the Food and Drug Administration (FDA) before its introduction into the market. The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table. Let μ1 be the true mean weight of individuals before starting the diet and let μ2 be the true mean weight of individuals after 3 weeks on the diet. Person Weight Before Diet Weight After Diet 1 166 159 2 211 206 3 204 201 4 213 207 5 220 216 Summary information is as follows: $\bar { d } = 5 , s _ { d } = 1.58$ . Calculate a 90% confidence interval for the difference between the mean weights before and after the diet is used.

Calculate the degrees of freedom associated with a small-sample test of hypothesis for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ , assuming $\sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 }$ and $n _ { 1 } = 13 , n _ { 2 } = 12 , s _ { 1 } = 1.3 , s _ { 2 } = 1.5$ .

A certain manufacturer is interested in evaluating two alternative manufacturing plans consisting of different machine layouts. Because of union rules, hours of operation vary greatly for this particular manufacturer from one day to the next. Twenty-eight random working days were selected and each plan was monitored and the number of items produced each day was recorded. Some of the collected data is shown below: DAY PLAN 1 OUTPUT PLAN 2 OUTPUT 1 1234 units 1311 units 2 1355 units 1366 units 3 1300 units 1289 units What type of analysis will best allow the manufacturer to determine which plan is more effective?

Which of the following represents the ratio of variances? A) $\frac { \sigma _ { 1 } { } ^ { 2 } } { \sigma _ { 2 } ^ { 2 } }$ B) $\frac { \mu _ { 1 } } { \mu _ { 2 } }$ C) $\frac { p _ { 1 } } { p _ { 2 } }$ D) $p _ { 1 } - p _ { 2 }$

Independent random samples from normal populations produced the results shown below. Sample 1: 5.8, 5.1, 3.9, 4.5, 5.4 Sample 2: 4.4, 6.1, 5.2, 5.7 a. Calculate the pooled estimator of $\sigma ^ { 2 }$ . b. Test $\mu _ { 1 } < \mu _ { 2 }$ using $\alpha = .10$ . c. Find a $90 \%$ confidence interval for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ .

A government housing agency is comparing home ownership rates among several immigrant groups. In a sample of 235 families who emigrated to the U.S. from Eastern Europe five years ago, 165 now own homes. In a sample of 195 families who emigrated to the U.S. from Pacific islands five years ago, 125 now own homes. Write a 95% confidence interval for the difference in home ownership rates between the two groups. Based on the confidence interval, can you conclude that there is a significant difference in home ownership rates in the two groups of immigrants?

In a controlled laboratory environment, a random sample of 10 adults and a random sample of 10 children were tested by a psychologist to determine the room temperature that each person finds most comfortable. The data are summarized below: Sample Mean Sample Variance Adults (1) 77. 4.5 Children (2) 74. 2.5 If the psychologist wished to test the hypothesis that children prefer warmer room temperatures than adults, which set of hypotheses would he use? A) $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) = 0$ vs. $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) < 0$ B) $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) = 3$ vs. $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) \neq 0$ C) $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) = 0$ vs. $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) > 0$ D) $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) = 0$ vs. $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) = 0$