a random sample of 20 prices for summer sleeping bags was taken with a sample mean $83.75 and a sample standard deviation of $29.97
a) find a 95% confidence interval for the mean price of all summer sleeping bags
b)what is m i.e. what is the margin f error?
To find the confidence interval for the mean price of all summer sleeping bags, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √n)
a) Finding the 95% confidence interval:
Since the sample size is 20, the degrees of freedom for a 95% confidence level is 19 (n - 1). Using a t-distribution table or a statistical software, we find that the critical value for a 95% confidence level and 19 degrees of freedom is approximately 2.093.
Plugging in the known values:
95% Confidence Interval = $83.75 ± (2.093) * ($29.97 / √20)
Calculating:
95% Confidence Interval = $83.75 ± (2.093) * $6.702
Simplifying:
95% Confidence Interval = $83.75 ± $14.02
The 95% Confidence Interval for the mean price of all summer sleeping bags is approximately $69.73 to $97.77.
b) Calculating the margin of error (m):
The margin of error can be found by taking half of the width of the confidence interval.
Margin of Error = (Upper Limit - Lower Limit) / 2
Substituting the values from the confidence interval:
Margin of Error = ($97.77 - $69.73) / 2
Calculating:
Margin of Error = $14.02 / 2
The margin of error (m) is approximately $7.01.