Question

Asked Apr 2, 2019

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Step 1

**Question 1:**

**Conditions:**

The conditions for the two proportion test for sample sizes *n*_{1 }and *n*_{2} with sample proportion for the two samples being *p*_{1 }and *p*_{2 }respectively, are:

- The data should be selected randomly.
- The products
*n*_{1}*p*_{1 }and*n*_{2}*p*_{2}, that is, the number of “successful cases” are greater than or equal to 10 for each sample. - The products
*n*_{1 }(1–*p*_{1}) and*n*_{2}(1–*p*_{2}), that is, the number of “failures” are greater than or equal to 10 for each sample.

**Conditions for this problem:**

It was found that 16 parts from alloy 1 did not pass the test out of 110 parts made and 14 parts from alloy 2 did not pass the test out of 46 parts made. It is checked whether the proportion of non-passing parts from alloy 1 is at least as large as the proportion of non-passing parts from alloy 2. The level of significance for the test is α=0.01.

Denote *p*_{1} and *p*_{2} as the proportion of parts of Alloy 1 that did not pass the test and proportion of parts of Alloy 2 that did not pass the test.

Therefore *n*_{1 }*= *110*, p*_{1 }*= *16/110 ≈ 0.145 and* n*_{2} *= *46*, p*_{2 }*= *14/46 ≈ 0.304.

For alloy 1, number of parts that did not pass the test is 16 > 10.

For alloy 2, number of parts that did not pass the test is 14 > 10.

Hence, the conditions satisfied. Therefore, we can use two-proportion *z*-test.

**Hypotheses:**

The aim is to check whether the proportion of non-passing parts from alloy 1 is at least as large as the proportion of non-passing parts from alloy 2.

The null hypothesis is:

*H*_{0}: *p*_{1} – *p*_{2 }< 0, that is, the proportion of non-passing parts from alloy 1 is less than the proportion of non-passing parts from alloy 2.

The alternative hypothesis is:

*H*_{1}: *p*_{1} – *p*_{2 }≥ 0, that is, the proportion of non-passing parts from alloy 1 is at least as large as the proportion of non-passing parts from alloy 2.

Step 2

Part A:

The test is a one tailed test,

**Decision rule:**

- If the test statistic is greater that the critical value (
*z*), reject the null hypothesis._{α} - Otherwise fail to reject the null hypothesis.

Step 3

Part B:

**Test statistic:**

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