Question

A random sample of 36 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 8 and the sample standard deviation is 2. Use a level of significance of 0.05 to conduct a two-tailed test of the claim that the population mean is 7.5.

(a) Is it appropriate to use a Student's *t*
distribution? Explain.

Yes, because the *x* distribution is mound-shaped and
symmetric and *σ* is unknown.

No, the *x* distribution is skewed
left.

No, the *x* distribution is skewed right.

No, the *x* distribution is not symmetric. No, *σ*
is known.

How many degrees of freedom do we use?

(b) What are the hypotheses?

*H*_{0}: *μ* < 7.5;
*H*_{1}: *μ* = 7.5

*H*_{0}: *μ* > 7.5;
*H*_{1}: *μ* =
7.5

*H*_{0}: *μ* = 7.5;
*H*_{1}: *μ* > 7.5

*H*_{0}: *μ* = 7.5;
*H*_{1}: *μ* < 7.5

*H*_{0}: *μ* = 7.5;
*H*_{1}: *μ* ≠ 7.5

(c) Compute the *t* value of the sample test statistic.
(Round your answer to three decimal places.)

* t* =

(d) Estimate the *P*-value for the test.

*P*-value > 0.250

0.100 < *P*-value <
0.250

0.050 < *P*-value < 0.100

0.010 < *P*-value < 0.050

*P*-value < 0.010

(e) Do we reject or fail to reject *H*_{0}?

At the *α* = 0.05 level, we reject the null hypothesis
and conclude the data are statistically significant.

At the *α* = 0.05 level, we reject the null hypothesis
and conclude the data are not statistically significant.

At the *α* = 0.05 level, we fail to reject the null
hypothesis and conclude the data are statistically significant.

At the *α* = 0.05 level, we fail to reject the null
hypothesis and conclude the data are not statistically
significant.

(f) Interpret the results.

There is sufficient evidence at the 0.05 level to reject the null hypothesis.

There is insufficient evidence at the 0.05 level to reject the null hypothesis.

Answer #1

Part a)

Yes, because the *x* distribution is mound-shaped and
symmetric and *σ* is unknown.

Part b)

*H*_{0}: *μ* = 7.5;
*H*_{1}: *μ* ≠ 7.5

Part c)

Test Statistic :-

t = ( X̅ - µ ) / ( S / √(n))

t = ( 8 - 7.5 ) / ( 2 / √(36) )

**t = 1.500**

Part d)

**P - value = P ( t > 1.5 ) = 0.1426**

Looking for value t = 1.500 in t table across n - 1 = 36 - 1 = 35
degree of freedom.

Part e)

Decision based on P value

P - value = P ( t > 1.5 ) = 0.1426

Reject null hypothesis if P value < α = 0.05 level of
significance

P - value = 0.1426 > 0.05 ,hence we fail to reject null
hypothesis

Conclusion :- Fail to reject null hypothesis

At the *α* = 0.05 level, we fail to reject the null
hypothesis and conclude the data are statistically significant.

There is insufficient evidence at the 0.05 level to reject the null hypothesis.

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