ten randomly selected people were asked how long they slept at night. The mean time was 7.1 hours and the (sample) standard deviation was 0.78 hours. Find the 95% confidence interval of the mean time. Assume the variable is normally distributed.
95% confidence interval indicates that you have 5% outside the confidence interval and 2.5% on either side. This is an area of .025. You can use a calculator or a t-table to find the critical value.
Multiply that value times the standard deviation.
add this value to the mean and then subtract this value to the mean which will give you the confidence interval. Always write the confidence interval with the smaller value first.
To find the 95% confidence interval of the mean time, we can use the formula:
Confidence interval = Mean ± (Z * Standard Error)
Where:
Mean is the sample mean time (7.1 hours)
Z is the critical value from the standard normal distribution corresponding to the desired confidence level (95% confidence level corresponds to Z = 1.96)
Standard Error is the standard deviation divided by the square root of the sample size (√10 in this case, since there are 10 randomly selected people)
Calculating the confidence interval:
Standard Error = 0.78 / √10 = 0.246
Confidence interval = 7.1 ± (1.96 * 0.246)
Confidence interval = 7.1 ± 0.482
Confidence interval = (6.618, 7.582)
Therefore, the 95% confidence interval for the mean time is (6.618 hours, 7.582 hours).
To find the 95% confidence interval of the mean time, we can use the formula:
Confidence interval = mean ± (critical value) x (standard deviation / √sample size)
Step 1: Calculate the critical value
Since the sample is small (less than 30) and the standard deviation is known, we will use the t-distribution to find the critical value. The critical value can be found using a t-table or a statistical software. For a 95% confidence level with 9 degrees of freedom (10 people - 1), the critical value is approximately 2.262.
Step 2: Calculate the confidence interval
Using the formula, we have:
Confidence interval = 7.1 ± (2.262) x (0.78 / √10)
Confidence interval = 7.1 ± (2.262) x (0.246)
Confidence interval = 7.1 ± 0.557
Step 3: Simplify the confidence interval
The lower limit of the confidence interval is given by 7.1 - 0.557 = 6.543 hours.
The upper limit of the confidence interval is given by 7.1 + 0.557 = 7.657 hours.
Therefore, the 95% confidence interval of the mean time is approximately 6.543 hours to 7.657 hours.