For a population with a standard deviation of σ = 20 | Complete Solution

1.         For a population with a standard deviation of σ = 20, how large a sample is necessary to

            have a standard error that is:

            a.         less than or equal to 10 point?

            b.         less than or equal to 4 points?

            c.         less than or equal to 2 point?

2.         The following samples were obtained from a population with a mean of μ = 40 and a standard deviation of σ = 8. Find the z-score corresponding to each sample mean.

            a.         A sample of n = 4 scores with M = 38

            b.         A sample of n = 16 scores with M = 38

            c.         A sample of n = 64 scores with M = 38

3.         Jumbo shrimp are those that require 10 to 15 shrimp to make a pound. Suppose that the 
            number of jumbo shrimp in a 1-pound bag averages μ = 12.5 with a standard deviation of
            σ =1, and forms a normal distribution. What is the probability of randomly picking a
            sample on n=25 1-pound bags that average more than M=13 shrimp per bag?

4.         A population forms a normal distribution with a mean of μ = 75 and a standard deviation of σ = 20.

a.         What proportion of the sample means for n = 25 have values less than 80? In other words, find p(M < 80) for n = 25.

b.         What proportion of the sample means for n = 100 have values greater than 79? In other words, find p(M > 79) for n = 100.

5.         For a normal-shaped distribution with μ = 50 and σ = 8:

            a.         What proportion of the scores has values between 46 and 54?

b.         For samples of n = 4, what proportion of the sample mean have values between 46 and 54?

c.         For samples of n = 16, what proportion of the sample mean have values between 46 and 54?

6.         A sample of n = 16 individuals is selected from a population with a mean of μ = 78. A treatment is administered to the individuals in the sample and, after treatment, the sample variance is found to be σ ² = 144.

a.         If the treatment produces a sample mean of M = 82, is this sufficient to conclude that there is a significant treatment effect using a two-tailed test with α = .05

b.         If the treatment produces a sample mean of M = 86, is this sufficient to conclude that there is a significant treatment effect using a two-tailed test with α = .05?

7.         The herbal supplement ginko biloba is advertised as producing an increase in physical strength and stamina. To test this claim, a sample of n = 36 adults is obtained and each person is instructed to take the regular daily dose of the herb for a period of 30 days. At the end of the 30-day period, each person is tested on a standard treadmill task for which the average, age-adjusted score is μ = 15. The individuals in the sample produce a mean score of M = 16.9 with SS = 1260.

a.         Are these data sufficient to conclude that the herb has a statistically significant effect using a two-tailed test with α = .05?

b.         What decision would be made if the researcher used a one-tailed test with α = .05? (Assume that the herb is expected to increase scores.)

8.         In a study examining the effects of alcohol on reaction time, Liguori and Robinson

            (2001) found that even moderate alcohol consumption significantly slowed response time

            to an emergency situation in a driving simulation. In a similar study, researchers

            measured reaction time 30 minutes after participants consumed one 6-ounce glass of

            wine. Again, they used a standardized driving simulation task for which the regular

            population averages μ = 400 msec. The distribution of reaction times is approximately

 normal with σ = 40. Assume that the researcher obtained a sample mean of M = 422 for

the n = 25 participants in the study.

            a.         Are the data sufficient to conclude that the alcohol has a significant effect on

                        reaction time? Use a two-tailed test with α = .01.

            b.         Do the data provide evidence that the alcohol significantly increased (slowed)

                        reaction time? Use a one-tailed test with α = .05.

9.         A random sample is obtained from a population with a mean of μ = 100, and a treatment is administered to the sample. After treatment, the sample mean is found to be M = 104 and the sample variance is σ ² = 400.

a.                   Assuming the sample contained n = 16 individuals, measure the size of the treatment effect by computing the Cohen’s d.

b.                  Assuming the sample contained n = 25 individuals, measure the size of the treatment effect by computing the Cohen’s d.

10.       Numerous studies have shown that IQ scores have been increasing, generation by generation, for years (Flynn, 1984, 1999). The increase is called the Flynn Effect, and the data indicate that the increase appears to be about 7 points per decade. To demonstrate this phenomenon, a researcher obtains an IQ test that was written in 1980. At the time the test was prepared, it was standardized to produce a population mean of μ  = 100. The researcher administers the test to a random sample of n = 16 of today’s high school students and obtains a sample mean IQ of M = 121 with SS = 6000. Is this result sufficiently higher than would be expected from a population with μ  = 100? Use a one-tailed test with α = .01.

Extra Credit (1 point)

            A researcher is investigating the effectiveness of a new study-skills training program for

            elementary school children. A sample of n = 64 third-grade children is selected to

            participate in the program and each child is given a standardized achievement test at the

            end of the year. For the regular population of third-grade children, scores on the test form

a normal distribution with a mean of µ = 150 and a standard deviation of σ = 22. The

mean for the sample is M = 156.

            a.         Identify the independent and the dependent variables for this study.

            b.         Assuming a two-tailed test, state the null hypothesis in a sentence that includes

                        the independent variable and the dependent variable.

c.         Using symbols, state the hypotheses (H₀ and H₁) for the two-tailed test.

d.         Calculate the test statistic (z-score) for the sample.