# Hypotheses

1. A study compared the step length in meters of amputees using a solid ankle cushion heel (SACH) prosthesis with those using a Carbon Copy II (CCII) device. For the random sample of 10 using SACH, the mean step length was 0.764 meter, and the standard deviation was 0.0011 meter. For the sample of 10 using CCII, the mean step length was 0.766 meter and the standard deviation was 0.00054 meter. Test the hypothesis at the 0.01 level that there is no difference in the population mean step length for the SACH and the CCII.

2. Transcutaneous electrical nerve stimulation (TENS) devices are frequently used in the management of acute and chronic pain conditions. An important component of the TENS system is the skin electrode. A study was conducted to determine conductive differences among the electrodes used with TENS devices. A sample of 11 electrodes from a low impedance group was tested in two different trials. Results follow

Electrode Trial 1 Impedance Trial 2 Impedance

1 1200 1900

2 1200 1100

3 1000 1000

4 1600 1600

5 1400 1600

6 1400 1400

7 1200 1100

8 1700 1400

9 1600 1800

10 1300 1400

11 1600 1400

Use the paired-t procedure to test the hypothesis at the 0.05 level, that there is no significant difference between the impedance measurements in the two trials.

3. Doctors conducted a study on the effects of folic acid on birth defects. One group of 2701 women took vitamins containing 0.8 mg of folic acid daily. The other group of 2052 women received only trace elements of folic acid. Among the group taking the folic acid, 35 cases of major birth defects developed. Among the group not taking folic acid, there were 47 cases of major birth defects. At the 0.01 level of significance, do the data provide sufficient evidence to conclude that the percentage of birth defects is lower in women who take folic acid?

#### Solution Preview

Please refer to the attachment.

1. A study compared the step length in meters of amputees using a solid ankle cushion heel (SACH) prosthesis with those using a Carbon Copy II (CCII) device. For the random sample of 10 using SACH, the mean step length was 0.764 meter, and the standard deviation was 0.0011 meter. For the sample of 10 using CCII, the mean step length was 0.766 meter and the standard deviation was 0.00054 meter. Test the hypothesis at the 0.01 level that there is no difference in the population mean step length for the SACH and the CCII.

Ho: µ1 - µ2 = 0, or Md=0

H1: µ1 - µ2 <> 0 or Md<>0

where Md is the difference between sample means, µ1 - µ2 is the difference between population means specified by the null hypothesis (usually zero), and is the estimated standard error of the difference between means.

Here, Md=µ2-µ1= 0.766 - 0.764= 0.002

The estimated standard error, , is computed assuming that the variances in the two populations are equal. The two sample sizes are equal (n1 = n2) then the population variance σ2 (it is the same in both populations) is estimated by using the following formula:

MSE = ( + )/2

where MSE (which stands for mean square error) is an estimate of σ2. Once MSE is calculated, can be ...

#### Solution Summary

1. A study compared the step length in meters of amputees using a solid ankle cushion heel (SACH) prosthesis with those using a Carbon Copy II (CCII) device. For the random sample of 10 using SACH, the mean step length was 0.764 meter, and the standard deviation was 0.0011 meter. For the sample of 10 using CCII, the mean step length was 0.766 meter and the standard deviation was 0.00054 meter. Test the hypothesis at the 0.01 level that there is no difference in the population mean step length for the SACH and the CCII.

2. Transcutaneous electrical nerve stimulation (TENS) devices are frequently used in the management of acute and chronic pain conditions. An important component of the TENS system is the skin electrode. A study was conducted to determine conductive differences among the electrodes used with TENS devices. A sample of 11 electrodes from a low impedance group was tested in two different trials. Results follow

Electrode Trial 1 Impedance Trial 2 Impedance

1 1200 1900

2 1200 1100

3 1000 1000

4 1600 1600

5 1400 1600

6 1400 1400

7 1200 1100

8 1700 1400

9 1600 1800

10 1300 1400

11 1600 1400

Use the paired-t procedure to test the hypothesis at the 0.05 level, that there is no significant difference between the impedance measurements in the two trials.

3. Doctors conducted a study on the effects of folic acid on birth defects. One group of 2701 women took vitamins containing 0.8 mg of folic acid daily. The other group of 2052 women received only trace elements of folic acid. Among the group taking the folic acid, 35 cases of major birth defects developed. Among the group not taking folic acid, there were 47 cases of major birth defects. At the 0.01 level of significance, do the data provide sufficient evidence to conclude that the percentage of birth defects is lower in women who take folic acid?

1. Cheese consumption in the United States was 28.0 pound per person in 1997. Figures for last year's cheese consumption are given below. At the .10 confidence level, has per capita cheese consumption in the U. S. increased since 1997. (Assume that σ = 6.9 pounds.)

40 23 27 32 36 34 28

27 26 30 22 41 20 36

30 26 39 18 33 22 27

38 27 20 31 21 25 30

31 31 30 16 38 30 23

2. The FBI reports that purse snatching in 1998 resulted in a mean loss per victim of $362. A random sample of twelve victims of purse snatching victims in the current year is surveyed. The sample mean loss is $314.10 and the sample standard deviation is $86.90. At the 0.05 level, can we assert that the mean loss for purse snatching victims has decreased since 1998?

3. According to Nielsen Media Research, in 1998, during the time slot from 8:00 PM to 11:00 PM, the average person watched 7 hours and 43 minutes of TV per week. A random sample of 40 women, age 18 - 24, reveals that their mean viewing time during the time slot from 8:00 PM to 11:00 PM was 343.25 minutes with a sample standard deviation of 222.2 minutes. At the 0.05 level, does the sample indicate that women in the 18 - 24 age group spend less time than the average person watching TV in this time slot?

4. The table that follows gives the distribution of the maximum wind speed of 1,293 tornadoes. Using a 0.01 level of significance, test the hypothesis that 50% of the population of tornadoes has wind speeds less than 72 mph.

Wind Speed Number

< 72 696

73 to 112 411

113 to 157 129

158 to 206 43

207 to 260 13

260 to 319 1

5. A computer systems engineer has researched data for e-mail messages handled by a particular server in a single weekday. The data from a sample of 22 randomly selected weekdays indicates that the sample mean number of e-mails per day was 41354.136. The sample standard deviation was 7912.235 e-mails per day. At the 0.05 level of confidence, does that data indicate that the population standard deviation is no greater than 5000 e-mails per weekday?