Exercises - Confidence Intervals and Hypothesis Tests for Means

  1. The braking distances of a simple random sample of cars has $n = 32$. and $\overline{x} = 132$ ft. Find the margin of error and 95% confidence interval for the braking distances of cars, if the necessary assumptions are met. Also, $\sigma$ is known to be 7 ft.

  2. A random sample of the birth weights of 186 babies has a mean of 3103 g and a standard deviation of 696 g. These babies are born to mothers who did not use cocaine during their pregnancy.

    1. What is the best point estimate of the mean weight of babies born to mothers who did not use cocaine during their pregnancies?

    2. Construct a 95% confidence interval estimate of the mean birth weight for all such babies.

    3. Compare the confidence interval from part b) to this confidence interval obtained for birth weights (in grams) of babies born to mothers who used cocaine during pregnancy: $2608 \lt \mu \lt 2792$ Does cocaine use appear to affect the birth weight of a baby?

  3. How many cars must be randomly selected and tested in order to estimate the mean braking distance of registered cars in the United States. We want 99% confidence that the sample mean is within 2 ft of the population mean and the population standard deviation is 7 ft.

  4. In a study designed to test the effectiveness of acupuncture for treating migraine, 142 subjects were treated with acupuncture and 80 subjects were given a sham treatment. The numbers of migraine attacks for the acupuncture treatment group had a mean of 1.8 and a standard deviation of 1.4. The numbers of migraine attacks for the sham treatment group had a mean of 1.6 and a standard deviation of 1.2.

    1. Construct the 95% confidence interval estimate of the mean number of migraine attacks for those treated with acupuncture.

    2. Construct the 95% confidence interval estimate of the mean number of migraine attacks for those given a sham treatment.

    3. Compare the two confidence intervals. What do the results suggest about the effectiveness of acupuncture?

  5. A simple random sample of 50 adults is obtained, and each person’s red blood cell count (in cells per microliter) is measured. The sample mean is 5.23. The population standard deviation for red blood cell counts is 0.54. Use a 0.01 significance level to test the claim that the sample is from a population with a mean less than 5.4, which is the value often used for the upper limit of the range of normal values. Use the P-value method.

  6. The average 1-ounce chocolate chip cookie contains 110 calories. A random sample of 15 different brands of cookies resulted in the following calorie amounts. At the 0.01 significance level, is there sufficient evidence that the average caloric content is greater than 110 calories? Use the critical value method.

    $$\begin{array}{cccccccc} 100 & 125 & 150 & 160 & 185 & 125 & 155 & 145\\ 160 & 100 & 150 & 140 & 135 & 120 & 110 & \end{array}$$
  7. In an analysis investigating the usefulness of pennies, the cents portions of 100 randomly selected credit card charges are recorded. The sample has a mean of 47.6 cents and a standard deviation of 33.5 cents. If the amounts from 0 to 99 cents are all equally likely, the mean is expected to be 49.5 cents. Use the confidence interval method at a 0.01 significance level to test the claim that the sample is from a population with a mean equal to 49.5 cents. What does the result suggest about the cents portions of credit card charges?

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