Question

The quality control team wish to investigate the rollover rates between two types of vehicles, A and B. Among 1200 tests on each type, 39 of the ‘A’s and 33 of the ‘B’s failed the test. Can we conclude that both vehicles have same rollover rates? Assume ? = 0.05.

a. Write the appropriate hypothesis.

b. Use P-value approach for hypothesis testing.

c. Use z-test for hypothesis testing.

d. Use confidence interval for hypothesis testing.

e. Clearly write your conclusion.

f. If the true rollover rates are ??′ = 0.035 and ??′ = 0.025, what is the type II error?

g. If the true rollover rates are ??′ = 0.035 and ??′ = 0.025, what is the minimum sample size needed to detect that difference with 99% probability?

Answer 1

For sample 1[A's rollover], we have that the sample size is N_1= 1200,

the number of cases is X1​= 39,

so then the sample proportion is

For sample 2 [ B's Rollover], we have that the sample size is N_2 = 1200,

the number of cases is X_2 = 33,

so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p_1 = p_2​

Ha:

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

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

z_c = 1.96

The rejection region for this two-tailed test is

R={z:∣z∣>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that

∣z∣=0.718 ≤ zc​ =1.96,

it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is p = 0.4728,

and since p =0.4728≥0.05,

it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected.

Therefore, there is not enough evidence to claim that the population proportion p_1​[A] is different than p_2[B], at the 0.05 significance level.

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