Question

The following hypotheses are given.

H₀ : π ≤ 0.70
H₁ : π > 0.70

A sample of 100 observations revealed that p = 0.75. At the 0.05 significance level, can the null hypothesis be rejected?

1. State the decision rule. (Round your answer to 2 decimal places.)

2. Compute the value of the test statistic. (Round your answer to 2 decimal places.)​​​​​​​

3. What is your decision regarding the null hypothesis?

• Reject H₀

• Do not reject H₀

Answer 1

Answer :

The following information is provided: The sample size is n=100. p= 0.75. the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H₀ : π ≤ 0.70
H₁ : π > 0.70

This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is z_c =1.64.

The rejection region for this right-tailed test is R= {z : z > 1.64}

(3) Test Statistics

The z-statistic is computed as follows:

z= ( p̂ - p₀ )/√[(p₀*(1 - p₀))/n]

Where,

p̂ = sample proportion

p₀ = Hypothesized proportion

n = sample size

Here, p̂ = p = 0.75, p₀ = π = 0.70 , n= 100

Therefore,

z = ( 0.75 - 0.70 ) / √[0.70*(1 - 0.70)/100]

z = 0.05 / √[0.21/100]

z = 0.05/0.0458

z = 1.0917

z ≈ 1.09

The value of the test statistic is 1.09.

(4) Decision about the null hypothesis :

Since it is observed that z=1.09 ≤ z_c =1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value for z score 1.09 at 5% level of significance for right tailed is p = 0.1378, and since p=0.1378 ≥ 0.05, it is concluded that the null hypothesis is not rejected.

That is, decision regarding the null hypothesis is,

Do not reject H₀.

(5) Conclusion:

It is concluded that the null hypothesis H₀ is not rejected. Therefore, there is not enough evidence to claim that the population proportion π is greater than p₀ , at the α=0.05 significance level.