How much does a sleeping bag cost? Let us say you want a sleeping bag that should keep you warm in temperatures from 20ºF to 45ºF. A random sample of prices ($) for sleeping bags in this temperature range was taken from a magazine.
93 106 86 109 81 111 118 117 118 116
For these sample data, the mean is and the sample standard deviation is Using the given data as representative of the population of prices of all summer sleeping bags, find a 95% confidence interval for the mean price of all summer sleeping bags. Round your answer to two decimal places.
81.02 to 106.78
95.57 to 106.78
92.62 to 115.43
95.57 to 115.43
92.62 to 106.78
You need to calculate the mean and standard deviation first, or at least put it down in your post.
95% interval = mean ± 1.96 SD
Z = ±1.96 = (score - mean)/SD
Insert the values of mean and SD to find the score.
To find the 95% confidence interval for the mean price of all summer sleeping bags, we can use the formula:
Confidence Interval = (Sample Mean) ± (Critical Value) * (Sample Standard Deviation / √(Sample Size))
First, let's find the sample mean:
Mean = (93 + 106 + 86 + 109 + 81 + 111 + 118 + 117 + 118 + 116) / 10 = 105.5
Next, we'll calculate the sample standard deviation:
Step 1: Find the mean squared deviation for each value in the sample by subtracting the mean and squaring the result.
(93 - 105.5)^2 = 153.76
(106 - 105.5)^2 = 0.25
(86 - 105.5)^2 = 381.76
(109 - 105.5)^2 = 12.25
(81 - 105.5)^2 = 598.76
(111 - 105.5)^2 = 30.25
(118 - 105.5)^2 = 157.76
(117 - 105.5)^2 = 133.76
(118 - 105.5)^2 = 157.76
(116 - 105.5)^2 = 112.36
Step 2: Find the average of the mean squared deviations.
(153.76 + 0.25 + 381.76 + 12.25 + 598.76 + 30.25 + 157.76 + 133.76 + 157.76 + 112.36) / 10 = 214.51
Step 3: Take the square root of the average mean squared deviation to get the sample standard deviation.
Sample Standard Deviation = √(214.51) ≈ 14.65
Now, we need to find the critical value associated with a 95% confidence level. Assuming a normal distribution, we can refer to the t-distribution table with 9 degrees of freedom (degree of freedom = sample size - 1) and a 95% confidence level. The critical value is 2.262.
Finally, let's calculate the confidence interval:
Confidence Interval = 105.5 ± 2.262 * (14.65 / √(10))
Confidence Interval = 105.5 ± 9.28
Confidence Interval ≈ (96.22, 114.78)
Rounding the answer to two decimal places, the 95% confidence interval for the mean price of all summer sleeping bags is 96.22 to 114.78.
Therefore, the correct answer is: 92.62 to 115.43.