A statistician selected a random sample of 125 observations from a population with a known standard deviation equal to 16 and computed a sample mean equal to 77. (SHOW WORK for CREDIT; round to 3 decimal places) a) Estimate the population mean with 93% confidence. b) Repeat Part a) using a 89% level of confidence. c) Repeat Part a) using a population variance equal to 441. d) Repeat Part a) using a sample size equal to 625. Any changes to C.I.?

Given that a statistician selected a random sample of 125 observations from a population with a known standard deviation equal to 16 and computed a sample mean equal to 77

We can use Z critical values since population std deviation is known. Also sample size >30

We find std error of mean = [tex]\frac{16}{\sqrt{125} } \\=1.4311[/tex]

Margin of error = Z critical value * 1.4311

Z critical values for 93% = 1.81

for 89% = 1.60

n Std error Z critical Conf interval

89%

125 1.4311 1.6 (74.71024 79.28976 )

93%

1.81 (74.409709 79.590291 )

We find that when confidence level increases interval width increses.

c) When sigma changes to 441, std error changes to [tex]\frac{16}{\sqrt{441} } \\=0.7619[/tex]

So we get

n Std error Z critical Conf interval

89%

441 0.7619 1.6 (75.781 78.219 )

93%

1.81 (75.621 78.37904)

d) When n = 625, std error changes to 16/25 = 0.64

n Std error Z critical Conf interval

89%

441 0.64 1.6 (75.976 78.024 )

93%

1.81 (75.842 78.1584)

When sample size increases, confidence interval width decreases.