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Solutions to Selected Normal Distribution and Confidence Interval Problems

Question 1

1. Shade the indicated area on the standard normal curve, then find the probability using your calculator or a standard normal table:

a. P(z < -1.56)

b. P( z > 2.06)

2. Adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15.

a. What IQ score represents the 95th percentile?

b. What IQ score represents the 50th percentile?

3. If exam scores are normally distributed with a mean of 35 and a standard deviation of 10, what percent of the scores is:

a. greater than 34?

b. smaller than 42?

c. Less than 28 or more than 34?

4.Find the z-scores for which 80% of the distribution’s area lies between -z and z

5. The scores for males on the critical reading portion of the SAT are normally distributed with a mean of 498 and a standard deviation of 116.

a. Find the probability that a randomly selected person scores higher than 700.

b. Find the probability that a randomly selected person scores less than 600.

c. Random samples of size n = 20 are drawn from the population of male critical reading SAT scores, and the mean of each sample is determined. Use the central limit theorem to find the mean and standard deviation of the sampling distribution of the sample means.

d. Find the probability that the mean SAT scores for a random sample of size 20 is less than 450.

6. According to a recent Survey, 93 out of 120 of Americans over the age of 25 have earned a high school diploma. What is the sampling distribution of the sample proportions? (Find the mean and the standard deviation for the sample proportion)

7. In a survey of 1022 US adults, 779 think the US should put more emphasis on producing energy from solar power.

a. Find the point estimate for the population proportion p of US adults who think that there should be more governmental emphasis on solar power. Round to 3 decimal places.

b. Construct a 90% confidence interval for the population proportion. Interpret your results.

8. Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people sampled, 421 responded yes – they own cell phones.

Using a 95% confidence level, compute a confidence interval estimate for the true proportion of adult residents of this city who have cell phones.

9. A manufacturer of candy must monitor the weight of candies in each bag. Bags that are too heavy cost the company money, and bags that are too light cause customers to be unhappy. A simple random sample of 40 bags of candy is selected, the mean weight is found to be 2.1 ounces and the standard deviation was 0.3 ounces.

Construct a 95% confidence interval for the population mean weight.

10. The data set represents the weights (in grams) of 10 randomly selected adult male fox squirrels from a forest. Assume weights are normally distributed.

821 957 782 930 720 821 794 976 810 941

a. Find the sample mean, and the sample standard deviation. Round both to one decimal place. (okay to use TI84 only to get the mean and standard deviation for the sample)

b. Construct a 99% confidence interval for the population mean. Interpret your results.