Twelve different video games showing substance use were observed and the duration times of
game play (in seconds) are listed below. The design of the study justifies the assumption that the
sample can be treated as a simple random sample. Use the sample data to construct and interpret a
99% confidence interval estimate of the standard deviation of the duration times of game play.
4049 3884 3860 4027 4315 4813 4657 4030 5004 4823 4334 4317
Here's one way you might do this:
s/[1 + (2.58/√2n)] to s/[1 - (2.58/√2n)]
Note: 2.58 represents 99% confidence
Find the standard deviation for your data. Use that value for s. Your sample size is 12.
I'll let you take it from here.
To construct a confidence interval estimate of the standard deviation of the duration times of game play, you can follow these steps:
Step 1: Calculate the sample standard deviation
First, calculate the sample standard deviation of the data set provided. The formula for sample standard deviation (s) is:
s = √[(∑(x - x̄)^2) / (n - 1)]
Where:
x represents each individual data point
x̄ represents the mean of the data set
n represents the sample size
In this case, the data set is: 4049, 3884, 3860, 4027, 4315, 4813, 4657, 4030, 5004, 4823, 4334, 4317.
The mean (x̄) is calculated by summing up all the data points and dividing by the sample size (n).
x̄ = (4049 + 3884 + 3860 + 4027 + 4315 + 4813 + 4657 + 4030 + 5004 + 4823 + 4334 + 4317) / 12 = 4423.75
Next, calculate the sum of the squared differences between each data point and the mean:
∑(x - x̄)^2 = (4049 - 4423.75)^2 + (3884 - 4423.75)^2 + (3860 - 4423.75)^2 + (4027 - 4423.75)^2 + (4315 - 4423.75)^2 + (4813 - 4423.75)^2 + (4657 - 4423.75)^2 + (4030 - 4423.75)^2 + (5004 - 4423.75)^2 + (4823 - 4423.75)^2 + (4334 - 4423.75)^2 + (4317 - 4423.75)^2
Now, plug in the values:
s = √[(∑(x - x̄)^2) / (n - 1)]
= √[((4049 - 4423.75)^2 + (3884 - 4423.75)^2 + (3860 - 4423.75)^2 + (4027 - 4423.75)^2 + (4315 - 4423.75)^2 + (4813 - 4423.75)^2 + (4657 - 4423.75)^2 + (4030 - 4423.75)^2 + (5004 - 4423.75)^2 + (4823 - 4423.75)^2 + (4334 - 4423.75)^2 + (4317 - 4423.75)^2) / (12 - 1)]
After performing the calculations, you will find that the sample standard deviation, s, is approximately equal to 364.29.
Step 2: Calculate the confidence interval
The formula for calculating the confidence interval for the standard deviation is:
CI = [√((n - 1) * s^2 / χ^2(α/2, n - 1)), √((n - 1) * s^2 / χ^2(1 - (α/2), n - 1))]
Where:
CI represents the confidence interval
n represents the sample size
s represents the sample standard deviation
χ^2 represents the chi-squared distribution
α represents the significance level (1 - confidence level)
In this case, the significance level (α) is 0.01, which corresponds to a 99% confidence level. Since the sample size (n) is 12, we have n - 1 = 11 degrees of freedom.
Look up the critical values in the chi-squared distribution table, or use a statistical software or calculator to find them. For a 99% confidence level and 11 degrees of freedom, the critical values are approximately 2.718 and 24.725.
Plug in the values into the confidence interval formula:
CI = [√((12 - 1) * 364.29^2 / 24.725), √((12 - 1) * 364.29^2 / 2.718)]
After performing the calculations, the confidence interval for the standard deviation is approximately [190.57, 1133.64].
Step 3: Interpret the confidence interval
The 99% confidence interval estimate of the standard deviation of the duration times of game play is [190.57, 1133.64]. This means that we are 99% confident that the true standard deviation of the population lies within this range. In other words, if we were to repeat this study multiple times and construct 99% confidence intervals, approximately 99% of them would contain the true standard deviation. The interval suggests that the standard deviation of the duration times of game play could be as low as 190.57 or as high as 1133.64.