A cell phone service provider has selected a random sample of 20 of its customers in an effort to estimate the mean number of minutes used per day. The results of the sample included a sample mean of 34.5 minutes and a sample standard deviation equal to 11.5 minutes. Based on this information, and using a 95 percent confidence level:
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.025) and its Z score.
95% = mean ± 1.96 SEm
SEm = SD/√n
gc g h36 2
The critical Value is t=2.093
Degrees of Freedom
n-1
To estimate the mean number of minutes used per day for all customers of the cell phone service provider, we can use a confidence interval. A confidence interval indicates the range within which we can reasonably expect the true population mean to fall.
To construct a 95 percent confidence interval, we need to determine the margin of error and then calculate the lower and upper bounds of the interval.
1. Margin of Error:
The margin of error represents the maximum allowable difference between the sample mean and the true population mean. It is calculated by multiplying the critical value (obtained from a t-table based on the desired confidence level) by the standard deviation divided by the square root of the sample size.
For a 95 percent confidence level, the critical value is obtained from a t-table with a degrees of freedom of n-1. In this case, the sample size is 20, so the degrees of freedom will be 19. The critical value for a 95 percent confidence level with 19 degrees of freedom is approximately 2.093.
Margin of Error = Critical Value * (Standard Deviation / √Sample Size)
= 2.093 * (11.5 / √20)
= 2.093 * (11.5 / 4.472)
≈ 5.39
2. Confidence Interval:
The lower and upper bounds of the confidence interval are calculated by subtracting and adding the margin of error from/to the sample mean, respectively.
Lower Bound = Sample Mean - Margin of Error
= 34.5 - 5.39
≈ 29.11
Upper Bound = Sample Mean + Margin of Error
= 34.5 + 5.39
≈ 39.89
Therefore, based on the sample information provided, we can say with 95 percent confidence that the true population mean number of minutes used per day falls within the range of approximately 29.11 to 39.89 minutes.