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 data to construct a 95% confidence interval estimate of mean duration of game play.

4061,3886,3865,4023,4309,4821,4645,4021,5024,4812,4331,4325

To construct a 95% confidence interval estimate of the mean duration of gameplay, we first need to calculate the sample mean and the standard error.

Step 1: Find the sample mean
Add up all the duration times of gameplay and then divide by the number of observations (12 in this case).
4061 + 3886 + 3865 + 4023 + 4309 + 4821 + 4645 + 4021 + 5024 + 4812 + 4331 + 4325 = 50313
Sample mean = 50313 / 12 = 4192.75

Step 2: Find the standard error of the mean
The standard error measures the variability or precision of the sample mean. It can be calculated using the formula:
Standard Error = (Standard Deviation / √n)

a. Determine the sample standard deviation.
To calculate the sample standard deviation, find the squared difference between each observation and the sample mean, then sum those values and divide by n - 1.
First, subtract the sample mean from each observation:
4061 - 4192.75 = -131.75
3886 - 4192.75 = -306.75
3865 - 4192.75 = -327.75
4023 - 4192.75 = -169.75
4309 - 4192.75 = 116.25
4821 - 4192.75 = 628.25
4645 - 4192.75 = 452.25
4021 - 4192.75 = -171.75
5024 - 4192.75 = 831.25
4812 - 4192.75 = 619.25
4331 - 4192.75 = 138.25
4325 - 4192.75 = 132.25

Next, square each of these differences:
(-131.75)^2 = 17392.56
(-306.75)^2 = 94053.06
(-327.75)^2 = 107469.06
(-169.75)^2 = 28817.56
(116.25)^2 = 13502.56
(628.25)^2 = 394930.56
(452.25)^2 = 204150.56
(-171.75)^2 = 29452.56
(831.25)^2 = 690896.56
(619.25)^2 = 383622.56
(138.25)^2 = 19111.56
(132.25)^2 = 17493.56

Then, sum up these squared differences:
17392.56 + 94053.06 + 107469.06 + 28817.56 + 13502.56 + 394930.56 + 204150.56 + 29452.56 + 690896.56 + 383622.56 + 19111.56 + 17493.56 = 2224119.28

Finally, divide this sum by n - 1 (11) to get the sample variance:
Sample Variance = 2224119.28 / 11 = 202192.66

b. Calculate the standard deviation by taking the square root of the sample variance:
Sample Standard Deviation = √202192.66 ≈ 449.78

Now that we have the sample mean (4192.75) and the sample standard deviation (449.78), we can proceed to construct the confidence interval.

Step 3: Construct the confidence interval
The formula to calculate the confidence interval estimate of the mean is:
CI = [sample mean - (critical value * standard error), sample mean + (critical value * standard error)]

a. Find the critical value for a 95% confidence level.
For a 95% confidence level, we need to find the critical value corresponding to a 2-sided confidence interval with a sample size of 12. Using a t-distribution table or calculator, the critical value for a 95% confidence level and 11 degrees of freedom is approximately 2.201.

b. Calculate the standard error.
Standard Error = 449.78 / √12 ≈ 129.95

c. Plug the values into the confidence interval formula.
CI = [4192.75 - (2.201 * 129.95), 4192.75 + (2.201 * 129.95)]

Simplifying the formula:
CI = [4192.75 - 286.05, 4192.75 + 286.05]

Finally, we have the confidence interval estimate of the mean:
CI ≈ [3906.70, 4480.80]

Therefore, we are 95% confident that the true mean duration of gameplay is between 3906.70 seconds and 4480.80 seconds.

To construct a 95% confidence interval estimate of the mean duration of game play, we can use the formula:

Confidence Interval = X̄ ± Z * (σ/√n)

Where:
X̄ = Sample mean
Z = Z-value for the desired confidence level (95% confidence level corresponds to a Z-value of 1.96)
σ = Population standard deviation (unknown in this case)
n = Sample size

First, let's calculate the necessary values:

Sample mean (X̄) = (4061 + 3886 + 3865 + 4023 + 4309 + 4821 + 4645 + 4021 + 5024 + 4812 + 4331 + 4325) / 12
= 48928 / 12
= 4077.33

Sample size (n) = 12

Now, let's find the margin of error:

Margin of Error = Z * (σ/√n)

Since the population standard deviation (σ) is unknown, we'll use the sample standard deviation (s) as an estimate:

Sample standard deviation (s) = √[Σ(xi - X̄)² / (n-1)]

Where xi is each individual duration and X̄ is the sample mean.

Calculating:

(xi - X̄)² = [ (4061 - 4077.33)² + (3886 - 4077.33)² + (3865 - 4077.33)² + (4023 - 4077.33)² + (4309 - 4077.33)² + (4821 - 4077.33)² + (4645 - 4077.33)² + (4021 - 4077.33)² + (5024 - 4077.33)² + (4812 - 4077.33)² + (4331 - 4077.33)² + (4325 - 4077.33)² ] / (12 - 1)

Using a calculator or spreadsheet, we can calculate:

(xi - X̄)² = [52970.11 + 32430.78 + 33302.78 + 10687.56 + 45743.84 + 349439.64 + 95673.78 + 25615.00 + 299849.44 + 34649.44 + 39002.11 + 33533.44] / 11
= 1512746.62 / 11
= 137522.42

Sample standard deviation (s) = √(137522.42 / 11)
≈ √12502.95
≈ 111.79

Now, let's calculate the margin of error:

Margin of Error = 1.96 * (111.79 / √12)
= 1.96 * (111.79 / √12)
≈ 60.84

Finally, we can construct the confidence interval:

Confidence Interval = 4077.33 ± 60.84
= (4016.49, 4138.17)

Therefore, the 95% confidence interval estimate of the mean duration of game play is approximately (4016.49, 4138.17) seconds.

Find the mean first = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

95% = mean ± 1.96 SEm

SEm = SD/√n

I'll let you do the calculations.