Chapter 13 Solutions

13.1

a) The 95% confidence interval for the percent of all adults who want to lose weight is (48, 54).

b) The phrase "95% confidence" means that if the same survey were conducted many times with different samples, 95% of the time the interval constructed should contain the true proportion of people who want to lose weight.

13.2 The student is not correct. A 95% confidence interval does not mean that 95% of the population falls within the interval. It does mean that the confidence interval for 95% of the samples will contain the value of the parameter.

13.3

a) The standard deviation is 60/31.6 = 1.90.

c) Two standard deviations, or 3.8 points.

e) 95% of all confidence intervals constructed in this manner will contain the true mean.

13.4 In order to have 97.5% fall within the central area, there will be (1-.975)/2 = 0.0125 in each tail. Looking inside of Table A to find 0.0125, you can see that this occurs for the z-score of -2.24. Critical values are generally taken to be positive and the distribution is symmetric, so the critical value will be 2.24.

13.5 x= (3.412 + 3.414 + 3.415)/3 = 3.4137 So a 95% confidence for the mean is 3.4137 ± 1.96 * .001/1.732, or (3.4126, 3.4148)

13.7

a) The 80% confidence interval for μ is (0.8404-1.282*0.0068/1.732, 0.8404+1.282*0.0068/1.732) or (.8354,.8454).

b) The 99.9% confidence interval for μ is (0.8404-3.291*0.0068/1.732, 0.8404+3.291*0.0068/1.732) or (.8274,.8533).

c) The increasing confidence level results in wider confidence intervals.

13.9

a) 22 ± 1.96 * 50/31.6, or (18.90, 25.10)

b) 22 ± 1.96 * 50/15.8, or (15.80, 28.20)

c) 22 ± 1.96 * 50/63.2, or (20.45, 23.55)

d) The margins of error for samples of size 250, 1000, and 4000 are 6.2, 3.10, 1.55, respectively. Increasing the sample size results in smaller margins of error.

13.11 For a desired margin of error of 5 points with 99% confidence, and standard deviation 15, the sample size needs to be at least (2.576*15/5)²=59.72. The smallest sample size that would achieve that margin of error is 60.

13.14

1	1248
2	22336789
3	03455
4	0

There do not appear to be any bad outliers. The data may be somewhat skew-right, but it is not extreme.

b) x=25.67. A 90% confidence interval for the mean healing rate is 25.67 ± 1.645*8/4.24, or (22.56, 28.77).

c) The 95% confidence interval will be wider. In order to have a greater chance of including the mean, the interval must be wider.

13.17

223.9	01
223.9	6688899
224.0	002
224.0	069
224.1	02

The data appear to be somewhat skew-right.

b) x= 224.0019. A 95% confidence interval for the mean measurement is 224.0019 ± 1.96* .06/4, or (223.973, 224.031).

13.18

a) A 99% confidence interval for the mean study time of all first year students is (137 - 2.576*65/16.4, 137+2.576*65/16.4) or (126.8, 147.2).

b) In order for this interval to be valid for all first year students, the sample must be representative of all first year students. This is best accomplished via a simple random sample (SRS).

13.20 If the 30000 value was not removed, the 99% confidence interval would be (248 - 2.576*65/16.4, 248+2.576*65/16.4) or (237.8, 258.2).

13.21 For a desired margin of error of 1 micrometer per hour and 90% confidence, the sample size should be at least (1.645 * 8/1)² = 173.2. So the smallest sample size that would result in the desired margin of error is 174.

13.25 A 90% confidence interval would have a margin of error less than ± 3%.

13.26 If the poll had interviewed 1500 persons rather than 1060, the margin of error for 95% confidence would be less than ± 3%.

13.27 Changing the size of the population should not change the margin of error of the results because the margin of error depends only on the sample size, the level of confidence, and the standard deviation of the population.