Question

Significance tests A test of H0: p = 0.65 against Ha: p < 0.65 has test statistic z = −1.78. (a) What conclusion would you draw at the 5% significance level? At the 1% level? (b) If the alternative hypothesis were Ha: p ≠ 0.65, what conclusion would you draw at the 5% significance level? At the 1% level?

Answer 1

Answer:

(a) At 5% significance level, reject H0

At 1% significance level, reject H0

(b) At 5% significance level, fail to reject H0

At 1% significance level, fail to reject H0

Step-by-step explanation:

(a) The test is a one tailed test

At 5% significance level, the critical value is 1.645

Conclusion: Reject H0 because the test statistic -1.78 is less than the critical value 1.645

At 1% significance level, the critical value is 2.326

Conclusion: Reject H0 because the test statistic -1.78 is less than the critical value 2.326

(b) The test is a two tailed test

At 5% significance level, the critical value is 1.96. The region of no rejection of H0 lies between -1.96 and 1.96

Conclusion: Fail to reject H0 because the test statistic -1.78 falls within -1.96 and 1.96

At 1% significance level, the critical value is 2.576. The region of no rejection of H0 lies between -2.576 and 2.576

Conclusion: Fail to reject H0 because the test statistic falls within -2.576 and 2.576

Answer 2

The conclusion that can be made from an hypothesis test depends on

the significance level and p-value.

Response:

(a) The conclusion at 5% is there is statistical evidence to suggest that p < 0.65

At 1% level; fail to reject H₀: p = 0.65, there is statistical evidence to suggest that p = 0.65

(b) With Hₐ ≠ 0.65, the conclusion at the 5% significance level is that there is sufficient statistical evidence that p = 0.65

At the 1% level, fail to reject H₀: p = 0.65,

Which is the method to draw conclusion from an hypothesis test?

The null hypothesis, H₀: p = 0.65

The alternative hypothesis, Hₐ: p < 0.65

The z-score is z = -1.78, which gives;

The p-value = 0.0375

(a) The significance level is 5%

Which gives, α = 0.05

Given that the p-value is less than the significant level, we have that

there is sufficient evidence against the null hypothesis, given that the

probability that the null hypothesis is correct is less than the significant

level of 5%.

Therefore, reject H₀, p = 0.65

• There is sufficient statistical evidence to suggest that the the p is less than 0.65, (p < 0.65)

However, at 1% significant level, α = 0.01, and the p-value, p = 0.0375 is

larger than the significance level.

• Therefore, we fail to reject the null hypothesis and there is sufficient statistical evidence to suggest that p = 0.65

(b) Hₐ: p ≠ 0.65

We have;

$\alpha = \dfrac{5 \%}{2} = 2.5 \% = \mathbf{0.025}$

Which gives;

The p-value (0.0375) is larger than the significant level, therefore, we

fail to reject the null hypothesis.

• There is sufficient statistical evidence to suggest that p = 0.65

At the 1% level of significance, we have;

$\alpha = \dfrac{1 \%}{2} = 0.5 \% = 0.005$

Which gives;

The p-value at z = -1.78 (p = 0.0375) is larger than the significant level

Therefore;

• There is sufficient evidence to suggest that p = 0.65

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