Question 1197672
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Answer: The number of paintings that teachers have is different from 15 paintings.


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Explanation:


{{{mu}}} = Greek letter mu, represents the population mean
More specifically, mu is the population mean of the number of paintings per teacher household.


Null Hypothesis: mu = 15
Alternative Hypothesis: {{{mu <> 15}}}
The "not equal" sign in the alternative says we have a two-tailed test.


The data set is
{15, 13, 15, 15, 14, 12, 12, 14, 14, 15, 14, 15, 16, 13, 13}
There are n = 15 items in this data set.


Add up all of the values in the data set to get 
15+13+15+15+14+12+12+14+14+15+14+15+16+13+13 = 210


Divide that sum by n = 15 to compute the mean
mean = (sum of values)/n
mean = (210)/15
mean = 14
This is the sample mean xbar. Its job is to estimate the population mean mu.


Use a graphing calculator, spreadsheet, or statistical software to compute the standard deviation of the data set. We want the sample standard deviation.
You should find the sample standard deviation is approximately s = 1.195229


Let's compute the standard error (SE)
SE = s/sqrt(n)
SE = 1.195229/sqrt(15)
SE = 0.3086068007936
SE = 0.308607
This value is approximate.


Now we can find the test statistic
t = (xbar - mu)/SE
t = (14 - 15)/0.308607
t = -3.24036719841092
t = -3.24


The sample size is n = 15
The degrees of freedom (df) is
df = n-1
df = 15-1
df = 14
Use a stats calculator, spreadsheet, or similar to find the p-value is roughly 0.0059
Since we're doing a two-tailed test, we find the area to the left of t = -3.24 and then double that result


The p-value being smaller than alpha = 0.05 tells us to reject the null and conclude that {{{mu <> 15}}} is the case.


Therefore we conclude that the number of paintings owned by teachers is different from 15 paintings.
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