You are given the sample mean and standard deviation of the population. Use this information to construct the displaystyle{90}%{quadtext{and}quad} {95

necessaryh 2021-01-05 Answered
You are given the sample mean and standard deviation of the population. Use this information to construct the​ 90%and95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals.
From a random sample of 66 ​dates, the mean record high daily temperature in a certain city has a mean of 85.69F. Assume the population standard deviation is 13.60F.
The​ 90% confidence interval is
The​ 95% confidence interval is
Which interval is​ wider?
Interpret the results.
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Expert Answer

Derrick
Answered 2021-01-06 Author has 94 answers
Step 1
Given:
This mean recorded temperature is: x=85.69F
The population's standard deviation is: σ=13.60F
The number of random samples is: n=66 dates.
Confidence level 90%:
The expression for the confidence level is,
12α=0.9
α=10.92
=0.05
Step 2
From the graph of the normal distribution, the confidence level is symmetric about the P axis. The expression to calculate the corresponding value of P is,
P=1α
=10.05
=0.95
From the z-table corresponding to P=0.95, the value of z is 1.65
The expression to calculate the interval is,
c.I=x±zσn
Substitute the values in the above expression.
c.l=85.69±1.65×13.6066
=85.69±2.762
=(85.692.762,85.692.762)
=(82.928,88.452)
Therefore, the interval for confidence leven 90% is from 82.928 to 88.452
Step 3
Confidence level 95%:
The expression for the confidence level is,
12α=0.95
α=10.952
=0.025
From the graph of the normal distribution, the confidence level is symmetric about the P axis. The expression to calculate the corresponding value of P is,
P=1α
=10.025
=0.975
From the z-table corresponding to P=0.975, the value of z is 1.96.
The expression to calculate the interval is,
c.l=x±zσn
Substitute the values in the above expression.
c.l85.69±1.96×13.6066
=85.69±3.281
=(85.693.281,85.693.281)
=(82.409,88.971)
Hence, the interval for confidence level 95% is from 82.409 to 88.971, and the interval for confidence level 95% is the widest one.
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