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Hypothesis Tests of a Single Population

1. Explain carefully the distribution between each of the following pairs of terms:

a) Null and alternative hypotheses

b) Simple and composite hypotheses

c) One-sided and two-sided alternatives

d) Type I and Type II errors

e) Significance level and power

2. During 2000 and 2001 many people in Europe objected to purchasing genetically modified food that was produced by farmers in the United States. The U.S. farmers argued that there was no scientific evidence to conclude that these products were not healthy. The Europeans argued that there still might be a problem with food.

a) State the null and alternative hypotheses from the perspective of the Europeans. b) State the null and alternative hypotheses from the perspective of the U.S. farmers.

3. Bank cash machine need to be stocked with enough cash to meet demand over an entire weekend. However, the bank will lose out on interest payments on any excess cash stocked into the cash machines. A particular bank believes that the mean withdrawal rate per transaction is normally distributed with a mean of $150 and a standard deviation of $50. Is there any evidence that the bank has got its calculations wrong, if a random sample of 36 customer transactions gives a mean sample of $160? State your null and alternative hypotheses.

4. A random sample is obtained from a population with varianceσ2=625, and the sample mean is computed. Test the null hypothesis H0:μ=100 versus the alternative hypothesis H1:μ>100 withα=0.05. compute the critical value xc and state your decision rule for the following options:

a) Sample size n=25

b) Sample size n=16

c) Sample size n=44

d) Sample size n=32

5. A random sample of n=25 is obtained from a population with varianceσ2, and the sample mean is computed. Test the null hypothesis H0:μ=100 versus the alternative hypothesis H1:μ>100 withα=0.05. compute the critical value xc and state your decision rule for the following options:

a) The population variance is σ2=225.

b) The population variance is σ2=900.

c) The population variance is σ2=400.

d) The population variance is σ2=600.

6. A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weight is known to be normal, with a standard deviation of 0.4 ounces. A random sample of 16 boxes yielded a sample mean weight of 15.84 ounces. Test at the 10% significance level the null hypothesis that the population mean weight is at least 16 ounces.

7. A company that receives shipments of batteries tests a random sample of nine of them before agreeing to taken a shipment. The company is concerned that the true mean life-time for all batteries in the shipment should be at least 50 hours. From past experience it is safe to conclude that the population distribution of lifetimes is normal with a standard deviation of 3 hours. For one particular shipment the mean life-time for a sample of nine batteries was 48.2 hours. Test at the 10% level the null hypothesis that the population mean life-time is at least 50 hours.

8. Test the hypotheses

H0:μ≤100

H1:μ>100

Using the random sample of n=25, a probability of Type I error equal to 0.05, and the following sample statistics: a) x=106;s=15

b) x=104;s=10

c) x=95;s=10

d) x=92;s=18

9. Test the hypotheses

H0:μ=100

H1:μ<100

Using the random sample of n=36, a probability of Type I error equal to 0.05, and the following sample statistics: a) x=106;s=15

b) x=104;s=10

c) x=95;s=10

d) x=92;s=18

10. A random sample of 1562 undergraduate enrolled in management ethics courses was asked to respond on a scale from 1( strong disagree) to 7 (strong agree) to this proposition: “Senior corporate executives are interested in social justice.” The sample mean response was 4.27, and the...