WS_6 (1)

MA-UA 2224 Worksheet #6 Fall 2023 1. A random sample of 64 bags of white cheddar popcorn weighed, on average, 5.23 ounces with a standard deviation of 0.24 ounce. Test the hypothesis that μ = 5 . 5 ounces against the alternative hypothesis, μ < 5 . 5 ounces, at the 0.05 level of signifi- cance. 2. The average height of females in the freshman class of a certain college has historically been 162.5 centimeters with a standard deviation of 6.9 centimeters. Is there reason to believe that there has been a change in the average height if a random sample of 50 females in the present freshman class has an average height of 165.2 centimeters? Use a P-value in your conclusion. Assume the standard deviation remains the same. 3. On a popular self-image test that results in normally distributed scores, the mean score for public-assistance recipients is expected to be 65. A random sample of 28 public- assistance recipients in Emerson County is given the test. They achieve a mean score of 62.1, and their scores have a standard deviation of 5.83. Do the Emerson County public-assistance recipients test differently, on average, than what is expected at the 0.02 level of significance? 4. A manufacturer has developed a new fishing line, which the company claims has a mean breaking strength of 15 kilograms with a standard deviation of 0 . 5 kilogram. To test the hypothesis that μ = 15 kilograms against the alternative that μ < 15 kilograms, a random sample of 50 lines will be tested. The critical region is defined to be x < 14 . 9 . (a) Find the probability of committing a type I error when H 0 is true. (b) Evaluate type II error for the alternatives μ = 14 . 8 and μ = 14 . 9 kilograms. 5. According to Chemical Engineering, an important property of fiber is its water ab- sorbency. The average percent absorbency of 25 randomly selected pieces of cotton fiber was found to be 20 with a standard deviation of 1 . 5. A random sample of 25 pieces of acetate yielded an average percent of 12 with a standard deviation of 1 . 25. Is there strong evidence that the population mean percent absorbency is significantly higher for cotton fiber than for acetate? Assume that the percent absorbency is approx- imately normally distributed and that the population variances in percent absorbency for the two fibers are the same. Use a significance level of 0 . 05. 6. A study was made to determine if the subject matter in a physics course is better understood when a lab constitutes part of the course. Students were randomly selected to participate in either a 3-semesterhour course without labs or a 4-semester-hour course with labs. In the section with labs, 11 students made an average grade of 85 with a standard deviation of 4.7, and in the section without labs, 17 students made an 1

average grade of 79 with a standard deviation of 6.1. Does this indicate adding labs increases students grades by more than 5 points on average? Use a P-value in your conclusion and assume the populations to be approximately normally distributed with equal variance. 7. In a study conducted by the Department of Human Nutrition and Foods at Virginia Tech, the following data were recorded on sorbic acid residuals, in parts per million, in ham immediately after dipping in a sorbate solution and after 60 days of storage. Assuming the populations are normally distributed, and the difference is also normally distributed. Is there sufficient evidence, at the 0.05 level of significance, to say that the length of storage influences sorbicacid residual concentrations? 8. In a study on the fertility of married women conducted by Martin O’Connell and Carolyn C. Rogers for the Census Bureau in 1979, two groups of childless wives aged 25 to 29 were selected at random, and each was asked if she eventually planned to have a child. One group was selected from among wives married less than two years and the other from among wives married five years. Suppose that 240 of the 300 wives married less than two years planned to have children some day compared to 288 of the 400 wives married five years. Can we conclude that the proportion of wives married less than two years who planned to have children is significantly higher than the proportion of wives married five years? Make use of a P-value. 9. Past data indicate that the amount of money contributed by the working residents of a large city to a volunteer rescue squad is a normal random variable with a standard deviation of $ 1.40. It has been suggested that the contributions to the rescue squad from just the employees of the sanitation department are much more variable. If the contributions of a random sample of 12 employees from the sanitation department have a standard deviation of $ 1.75, can we conclude at the 0.01 level of significance that the standard deviation of the contributions of all sanitation workers is greater than that of all workers living in the city? 10. Two types of instruments for measuring the amount of sulfur monoxide in the at- mosphere are being compared in an air-pollution experiment. Researchers wish to 2