Question 1201612: You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If convenient, use technology to construct the confidence intervals.
A random sample of 55 home theater systems has a mean price of $117.00. Assume the population standard deviation is $18.30.
Answer by math_tutor2020(3816)   (Show Source):
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Answers:
90% confidence interval is (112.94, 121.06)
95% confidence interval is (112.16, 121.84)
The 95% confidence interval is wider.
Further explanation is down below.
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At 90% confidence, the z critical value is roughly z = 1.645
Use a table like this
https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
to get that value. Look at the bottom row labeled "Z" and above the 90% confidence level.
A stats calculator can also compute this value.
xbar = 117.00 = sample mean
sigma = 18.30 = population standard deviation
n = 55 = sample size
Let's compute the margin of error
E = margin of error
E = z*sigma/sqrt(n)
E = 1.645*18.30/sqrt(55)
E = 4.05915511193244
E = 4.059155
Then,
L = lower boundary
L = xbar - E
L = 117 - 4.059155
L = 112.940845
L = 112.94
and
U = upper boundary
U = xbar + E
U = 117 + 4.059155
U = 121.059155
U = 121.06
The 90% confidence interval for the population mean is 112.94 < mu < 121.06
That is in the format L < mu < U
It can be condensed to the format (L, U) and we get (112.94, 121.06)
This second format is more common, but the drawback is that we don't know what population parameter we're estimating (unless further context is given).
Interpretation: We are 90% confident the population mean price of a home theater system is somewhere between $112.94 and $121.06
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We'll repeat the process but this time for a 95% confidence interval.
At 95% confidence, the z critical value is roughly z = 1.96
Refer to that table I mentioned earlier.
The other input values are the same as before.
E = margin of error
E = z*sigma/sqrt(n)
E = 1.96*18.30/sqrt(55)
E = 4.83644013336631
E = 4.836440
The margin of error is larger than previously calculated.
This will widen the confidence interval.
L = lower boundary
L = xbar - E
L = 117 - 4.836440
L = 112.16356
L = 112.16
and
U = upper boundary
U = xbar + E
U = 117 + 4.836440
U = 121.83644
U = 121.84
The 95% confidence interval can be expressed as 112.16 < mu < 121.84
Or can be expressed as (112.16, 121.84)
Interpretation: We are 95% confident the population mean price of a home theater system is somewhere between $112.16 and $121.84
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Let's compare the confidence intervals.
90% confidence interval: (112.94, 121.06)
95% confidence interval: (112.16, 121.84)
The 95% confidence interval is wider because the margin of error is larger.
The more confident we get, the wider the interval will become.
Imagine you are looking for a certain rare species of fish in the ocean.
To be more confident you caught the fish you want, you could increase the size of the net.
The size of the net is analogous to the width of the confidence interval.
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