5. Students in an introductory statistics class were asked to report the age of their mothers when they were born. Summary statistics include
Sample size: 36 students Sample mean: 29.643 years
Sample standard deviation: 4.564 years
a. Calculate the standard error of this sample mean.
b. Determine and interpret a 90% confidence interval for the mother’s mean age (at student’s birth) in the population of all students at this university.
c. How would a 99% confidence interval compare to the 90% interval in terms of its midpoint and half-width?
d. Would you expect 90% of the ages in the sample to be within the 90% confidence interval? Explain why or why not.
e. Even if the distribution of mothers’ ages were somewhat skewed, would this confidence interval procedure still be valid with these data? Explain why or why not.
a. The standard error of the sample mean is 4.564/sqrt(36) = 0.7607
b. yes The mother's mean age student birth lie in interval (28.358, 30.928)
a. To calculate the standard error of the sample mean, you need the formula:
Standard Error = Sample Standard Deviation / Square Root of Sample Size
In this case, the sample standard deviation is given as 4.564 years, and the sample size is 36. Plugging these values into the formula, we have:
Standard Error = 4.564 / sqrt(36)
Calculating this expression, we find that the standard error of the sample mean is 4.564 / 6 = 0.761.
b. To determine the 90% confidence interval for the mother's mean age, you need the formula:
Confidence Interval = Sample Mean ± (Critical Value)*(Standard Error)
The critical value corresponds to the level of confidence and the sample size. For a 90% confidence level with a sample size of 36, the critical value can be found using a t-distribution table or a statistical software. For simplicity, let's assume the critical value is 1.69. Plugging the values into the formula, we have:
Confidence Interval = 29.643 ± (1.69)*(0.761)
Calculating this expression, we find that the 90% confidence interval for the mother's mean age is (28.093, 31.193) years.
Interpretation: We are 90% confident that the true mean age of mothers (at student's birth) in the population of all students at this university falls within the range of 28.093 to 31.193 years.
c. A 99% confidence interval would have a larger critical value than a 90% confidence interval, resulting in a wider interval. The midpoint of the interval would remain the same since it is based on the sample mean. However, the half-width of the interval would be larger for the 99% confidence interval compared to the 90% confidence interval.
d. No, we would not expect 90% of the ages in the sample to be within the 90% confidence interval. The confidence interval estimates the range within which the true population mean is likely to fall, not the range within which individual sample values are likely to fall. The confidence interval represents the uncertainty associated with estimating the population parameter based on a sample.
e. Yes, the confidence interval procedure would still be valid even if the distribution of mothers' ages were somewhat skewed. The validity of the confidence interval procedure does not rely on the shape of the distribution. It is based on the assumptions of normality for the sampling distribution of the mean, which can be met even with skewed data when the sample size is sufficiently large. However, if the sample size were small, the skewness of the distribution might need to be taken into consideration when interpreting the results.