Question

Hypothesis Testing: 1. You are given the following information and then asked to do a hypothesis test with a 95% level of confidence to evaluate whether or not it is likely that the population mean is equal to 76 for a given variable. N=10,000 X-bar=74 s2=10,000 2. You are given the following information and then asked to do a hypothesis test with a 95% level of confidence to evaluate whether or not it is likely that the population mean is equal to 440 for a given variable. N=1500 X-bar=450 s2=10,000 3. Explain the differences in these Cls, and why they occur.
please help me answer questions 1,2, 3.

Answer 1

(1)

(i)

n = 10000

= 76

T

s = = 100

10000

$\alpha$ = 0.05

df = 10000 - 1 = 9999

From Table, critical values of t = $\pm$ 1.96

Confidence Interval:

txs 1.96 x 100 T 76 + 76 = 1.96 = (74.04,77.96) n 10000

So,

Answer is:

(74.04, 77.96)

(ii)

Since 76 is included in the Confidence Interval: (74.04, 77.96), we conclude:

It is likely that the population mean is equal to 76.

(2)

(i)

n = 1500

= 450

T

s = = 100

10000

$\alpha$ = 0.05

df = 10000 - 1 = 9999

From Table, critical values of t = $\pm$ 1.962

Confidence Interval:

$\bar{x}\pm \frac{t\times s}{\sqrt{n}}=450\pm \frac{1.962\times 100}{\sqrt{1500}}=450\pm 5.065=(444.935,455.065)$

So,

Answer is:

(444.935 455.065)

(ii)

Since 440 is not included in the Confidence Interval: (444.935 455.065), we conclude:

It is not likely that the population mean is equal to 440.

(3)

The 2 CIs are different because as Sample Size = n is decreased from n = 10000 to n =1500, the width of the Confidence Interval inceases.