Question 1193182
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Part (a)


The best point estimate of the population mean is the sample mean. It will be handy later to form the confidence intervals.
We can use the notation xbar to indicate the sample mean.


Answer: 7.2


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Part (b)


n = 50 = sample size
sigma = 2.1 = population standard deviation


At 95% confidence, the z critical value is roughly z = 1.960
You'll need to either memorize this value or have it on a reference table. A specialized calculator is also another option.


E = margin of error
E = z*sigma/sqrt(n)
E = 1.960*2.1/sqrt(50)
E = 0.58209030227277
E = 0.582090


L = lower bound of confidence interval
L = xbar - E
L = 7.2 - 0.582090
L = 6.61791
L = 6.62


U = upper bound of confidence interval
U = xbar + E
U = 7.2 + 0.582090
U = 7.78209
U = 7.78


Notice how the sample mean xbar = 7.2 is the center of the confidence interval.


Answer: (6.62, 7.78)


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Part (c)


We use the same idea as part (b)
This time we have z = 2.576 as the critical value for a 99% confidence interval.
Every other variable stays the same.


E = margin of error
E = z*sigma/sqrt(n)
E = 2.576*2.1/sqrt(50)
E = 0.76503296870134
E = 0.765033


L = lower bound of confidence interval
L = xbar - E
L = 7.2 - 0.765033
L = 6.434967
L = 6.43


U = upper bound of confidence interval
U = xbar + E
U = 7.2 + 0.765033
U = 7.965033
U = 7.97


The point estimate is the same as last time. The only thing that changed is the margin of error. 
The higher the confidence level, the larger the margin of error will be. This in turn creates a wider confidence interval.

 
It's like trying to catch an elusive fish. The wider the net (i.e. the wider the interval), the more confident we are in catching the fish. 


Answer: (6.43, 7.97)
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