Amanda S. answered • 11/09/15
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Hello, Jamie!
In order to solve this problem, we must first assume that the conditions of simple random sampling and a normal distribution are satisfied.
Xbar is our sample mean. It is the measure that is used to estimate the true population's mean, plus or minus a certain amount of error - the margin of error.
s in this case is our standard deviation of the sample. Because we are not given the population's standard deviation, which would be denoted by σ, we need to ask ourselves the following question:
Is our population size at least 20x larger than our sample?
The answer is, of course, yes. Our sample size is n = 40. There are approximately 3.8 million Americans... so we can safely assume our population size is sufficiently large enough to use the following equation:
SExbar = s / √(n)
This is the standard error is an estimate of the population's standard deviation, which is seldom known. Using this equation we find:
SExbar = 1.3
Now that we have the standard error, we need to find our critical value. Since we do not have σ and our sample is relatively small, we should use a t-score instead of a z-score.
To find the critical value we have to....
1) Find α. α = 1 - (confidence level/100)
α = 1 - (99/100)
α = 0.01
2) Find the degrees of freedom (df).
df = n - 1
df = 40 - 1
df = 39
3) Find the t-score using a t-table, which you can either find online or in the back of our textbook.
Looking at a two tailed t-table for confidence intervals with a df = 39 and α = 0.01, we get a t = 2.708.
Now, to compute the margin of error, we need to use the following equation:
ME = SExbar * critical value
ME = (1.3)(t)
ME = (1.3)(2.708)
ME = 3.52
Now that we have our margin of error, all that we have less is to state our confidence interval (xbar ± ME).
We are 99% confident that the American population watches a mean of 15 ± 3.52 hours per week.
