Exam 8: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

A new type of band has been developed for children who have to wear braces. The new bands are designed to be more comfortable, look better, and provide more rapid progress in realigning teeth. An experiment was conducted to compare the mean wearing time necessary to correct a specific type of misalignment between the old braces and the new bands. One hundred children were randomly assigned, 50 to each group. A summary of the data is shown in the table. Old Braces New Bands 410 days 380 days s 41 days 65 days How many patients would need to be sampled to estimate the difference in means to within 26 days with probability 99%?

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

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: $\mu _ { \mathrm { d } } > 9$ ? $\mathrm { n } _ { \mathrm { d } } = 27 , \alpha = 0.025$

Specify the appropriate rejection region for testing $\mathrm { H } _ { 0 } : \sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ in the given situation. - $\mathrm { H } _ { \mathrm { a } } : \sigma _ { 1 } ^ { 2 } > \sigma _ { 2 } ^ { 2 } ; \alpha = 0.01 , \mathrm { n } _ { 1 } = 10 , \mathrm { n } _ { 2 } = 21$

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 $\mu _ { \mathrm { A } } - \mu \mathrm { B }$ was calculated: (100, 2500). Which of the following inferences is correct?

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 1 147 140 2 192 187 3 185 182 4 194 188 5 201 197 Summary information is as follows: =5,=1.58 . Calculate a 90% confidence interval for the difference between the mean weights before and after the diet is used.

We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarkets and recording the price of a six-pack of each brand of cola at each supermarket. The data are shown in the following table: Price Supermarket Brand 1 Brand 2 Difference 1 \ 2.25 \ 2.30 \ -0.05 2 2.47 2.45 0.02 3 2.38 2.44 -0.06 4 2.27 2.29 -0.02 5 2.15 2.25 -0.10 6 2.25 2.25 0.00 7 2.36 2.42 -0.06 8 2.37 2.40 -0.03 =2.3125 =2.3500 =-0.0375 =0.1007 =0.0859 =0.0381 If the problem above represented a paired difference, what assumptions are needed for a confidence interval for the mean difference to be valid?

Independent random samples were selected from each of two normally distributed populations, $n _ { 1 } = 7$ from population 1 and $n _ { 2 } = 9$ from population 2 . The data are shown below. Population 1: 2.5 3.1 2.3 1.8 4.2 3.5 3.9 Population 2: 2.9 1.7 4.6 3.5 3.7 2.8 4.6 3.4 1.9 Find the test statistic for the test of $H _ { 0 } : \sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } 2$ against $H _ { \mathrm { a } } : \sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 }$ .

Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of the two procedures, random samples of surgical patients were obtained and the numbers of patients who showed no recurrence of the disease after a 1-year period were recorded. The data are shown in the table. n Number of Successes Procedure A 100 77 Procedure B 100 87 How large a sample would be necessary in order to estimate the difference in the true success rates to within .10 with 95% reliability?

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

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 $n _ { 1 } = n _ { 2 } = 12$ .

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: $\mathrm { n } = 10 , \quad$ 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.

Find F_.05 where $v _ { 1 } = 8 \text { and } v _ { 2 } = 11$

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 assumptions are necessary for the above test to be valid?

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 Find the standard error of the estimate for the difference in mean comfortable room temperatures between adults and children.

Given $v _ { 1 } = 9 \text { and } v _ { 2 } = 5 , \text { find } P ( F > 6.68 )$

Which supermarket has the lowest prices in town? All claim to be cheaper, but an independent agency recently was asked to investigate this question. The agency randomly selected 100 items common to each of two supermarkets (labeled A and B) and recorded the prices charged by each supermarket. The summary results are provided below: =2.09 =1.99 =.10 =0.22 =0.19 =.03 Assuming a matched pairs design, which of the following assumptions is necessary for a confidence interval for the mean difference to be valid?

Data were collected from CEOs in the consumer products industry and the CEOs in the telecommunication industry. The data were analyzed using a software package in order to compare mean salaries of CEOs in the two industries. HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM SALARY $X =$ Consumer Products $\mathrm { Y } =$ Telecommunications SAMPLE MEAN OF $X = 1761$ SAMPLE VARIANCE OF $X = \quad 3.97555$ E6 SAMPLE SIZE OF $X = 21$ SAMPLE MEAN OF $Y = 1093.5$ SAMPLE VARIANCE OF $Y = 103255$ SAMPLE SIZE OF $\mathrm { Y } = \quad 21$ MEAN X-MEAN $\mathrm { Y } = \quad 667.5$ test statistic $= \quad 1.47809$ D. F. $= 40$ P-VALUE $= \quad 0.147626$ P-VALUE $/ 2 = 0.0738131$ SD. ERROR $= 451.597$ Using $\alpha = .05$ , give the rejection region for a two-tailed test.

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 andfemale purchasers of the product. The sample data is shown here: Female: $\mathrm { n } = 31 , \quad$ sample mean $= 50.30 , \quad$ sample standard deviation $= 13.215$ Male: $\mathrm { n } = 21 , \quad$ sample mean $= 39.80 , \quad$ 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 paired difference experiment produced the following results. $n _ { d } = 40 , \bar { x } _ { 1 } = 18.4 , \bar { x } _ { 2 } = 19.7 , \bar { d } = - 1.3 , s _ { d } = 5$ Perform the appropriate test to determine whether there is sufficient evidence to conclude that $\mu _ { 1 } < \mu _ { 2 }$ using $\alpha = .10$ .