A school psychologist reports that the mean number of hours the students at this school sleep each night is 8 hours. The students believe the mean is not 8 hours. To find an estimate of the true mean, they select a random sample of 15 students from their school and ask how many hours each slept last night. A stemplot and boxplot of the numbers of hours are shown here. t he sample mean is 6.67 hours with a sample standard deviation of 1.88 hours.
a. Verify that the conditions for computing a confidence interval are met in this case.
b. Compute and interpret a 95% confidence interval for the mean number of hours the students at this school sleep each night.
A school psychologist reports that the mean number of hours the students at this school sleep each night is 8 hours. The students believe the mean is not 8 hours. To find an estimate of the true mean, they select a random sample of 15 students from their school and ask how many hours each slept last night. A stemplot and boxplot of the numbers of hours are shown here. T he sample mean is 6.67 hours with a sample standard deviation of 1.88 hours.
a. Verify that the conditions for computing a confidence interval are met in this case.
b. Compute and interpret a 95% confidence interval for the mean number of hours the students at this school sleep each night.
a. In order to compute a confidence interval, there are certain conditions that need to be met. These conditions are:
1. Random Sample: The sample of 15 students should have been randomly selected from the population of students at the school. This ensures that the sample is representative of the population.
2. Normality: The population distribution should be approximately normal or the sample size should be large enough (n ≥ 30). This condition is important because the formula used to calculate the confidence interval assumes a normal distribution. To check for approximate normality, we can examine the stemplot and boxplot provided.
b. To compute a 95% confidence interval for the mean number of hours the students at this school sleep each night, we can use the following formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation of the sample mean)
First, we need to find the critical value. Since the sample size is small (n < 30), we need to use a t-distribution and calculate the degrees of freedom.
Degrees of Freedom (df) = sample size - 1 = 15 - 1 = 14
Next, we need to find the critical value from the t-distribution corresponding to a 95% confidence level for a two-tailed test. Using a t-table or statistical software, the critical value can be found to be approximately 2.145.
Now we can calculate the confidence interval:
Confidence Interval = 6.67 ± (2.145) * (1.88 / √15)
Calculating this expression will give us the lower and upper bounds of the confidence interval.
Interpretation: The 95% confidence interval for the mean number of hours the students at this school sleep each night is [lower bound, upper bound]. This means that we are 95% confident that the true mean lies within this range.