A simple random sample of 50 female 14-year-olds is selected. The sample mean height of the girls is found to be 62 inches. Assume the height of 14-year-old girls is normally distributed with a standard deviation of 5 inches.
1. Based on these data, a 95% confidence interval for the true mean height of all
14-year-old girls is:
A) 62 +/- 1.96
B) 62 +/- 1.386
C) 62 +/- 1.645
D) 62 +/- 0.196
2. Which of the following correctly interprets the 95% confidence interval for the true
mean height of all 14-year-old girls?
A) We can be 95% confident that the sample mean height of 14-year-old girls is within
the confidence interval obtained.
B) If this study were to be repeated with a sample of the same size, there is a 0.95
probability that the sample mean height of 14-year-old girls would be in the interval
obtained.
C) We can be 95% confident that the population mean height of all 14-year-old girls is
within the interval obtained.
D) 95% of all 14-year-old girls have heights within the interval obtained.
1. 95% = mean ± 1.96SEm
SEm = SD/√n
2. C
kik
To calculate a confidence interval, we need to use the sample mean, the standard deviation, and the sample size. In this case, we know the following information:
- Sample mean height of the girls: 62 inches
- Standard deviation: 5 inches
- Sample size: 50
1. To find the 95% confidence interval for the true mean height of all 14-year-old girls, we can use the formula:
Confidence Interval = Sample Mean +/- Critical Value * (Standard Deviation / Square Root of Sample Size)
The critical value depends on the desired confidence level. For a 95% confidence level, the critical value is 1.96. Plugging in the values, we get:
Confidence Interval = 62 +/- 1.96 * (5 / sqrt(50))
Confidence Interval = 62 +/- 1.96 * (5 / 7.07)
Confidence Interval = 62 +/- 1.96 * 0.71
Confidence Interval = 62 +/- 1.39
So, the correct answer is (B) 62 +/- 1.386.
2. The correct interpretation of the 95% confidence interval for the true mean height of all 14-year-old girls is (C) We can be 95% confident that the population mean height of all 14-year-old girls is within the interval obtained.
This means that we can expect the population mean height of all 14-year-old girls to fall within the calculated confidence interval with a confidence level of 95%.