twelve different vTideo 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 a 95% confidence interval estimate of u (mew) ((population mean)) the mean duration of game play. 4054 3874 3862 4034 4327 4807 4654 4025 5000 4832 4340 4309 What is the confidence interval estimate of the population mean u? ____ Round to one decimal place as needed.

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.

To construct a confidence interval estimate of the population mean (u), we can use the following formula:

Confidence Interval = x̄ ± (t * (s/√n))

where:
x̄ is the sample mean
t is the critical value from the t-distribution for the desired level of confidence
s is the sample standard deviation
n is the sample size

To find the confidence interval, we need to calculate the sample mean (x̄), the sample standard deviation (s), and the critical value (t) for a 95% confidence level.

Step 1: Calculate the sample mean (x̄):

Add up all the game play durations and divide by the number of observations:
x̄ = (4054 + 3874 + 3862 + 4034 + 4327 + 4807 + 4654 + 4025 + 5000 + 4832 + 4340 + 4309) / 12
x̄ = 49,398 / 12
x̄ ≈ 4,116.5

Step 2: Calculate the sample standard deviation (s):

First, calculate the variance by subtracting the sample mean from each observation, squaring the difference, summing them up, and dividing by the sample size minus 1:
variance = [(4054 - 4116.5)^2 + (3874 - 4116.5)^2 + ... + (4309 - 4116.5)^2] / (12 - 1)

Next, take the square root of the variance to get the standard deviation:
s = √(variance)

Step 3: Find the critical value (t):

Since the sample size (n) is small (12) and the population standard deviation is unknown, we need to use the t-distribution. For a 95% confidence level with 11 degrees of freedom (n - 1), the critical value can be found using a t-distribution table or an online calculator.

Assuming a two-tailed test, the critical value (t) is approximately 2.201.

Step 4: Calculate the confidence interval:

Now that we have all the required values, substitute them into the confidence interval formula:

Confidence Interval = x̄ ± (t * (s/√n))
Confidence Interval = 4,116.5 ± (2.201 * (s/√12))

Round the confidence interval to one decimal place.