Question

In a study of the accuracy of fast food​ drive-through orders, one restaurant had 40 orders that were not accurate among 307 orders observed. Use a 0.05 significance level to test the claim that the rate of inaccurate orders is greater than ​10%. State the test result in terms of the claim. Identify the null and alternative hypotheses for this test The test statistic for this hypothesis test is? The​ P-value for this hypothesis test is? Identify the conclusion for this hypothesis test. State the test result in terms of the claim.

Answer 1

Answer:

We conclude that the rate of inaccurate orders is greater than ​10%.

Step-by-step explanation:

We are given that in a study of the accuracy of fast food​ drive-through orders, one restaurant had 40 orders that were not accurate among 307 orders observed.

Let p = population proportion rate of inaccurate orders

So, Null Hypothesis, $H_0$ : p $\leq$ 10%     {means that the rate of inaccurate orders is less than or equal to ​10%}

Alternate Hypothesis, $H_A$ : p > 10%      {means that the rate of inaccurate orders is greater than ​10%}

The test statistics that will be used here is One-sample z-test for proportions;

                          T.S.  =  $\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n} } }$  ~ N(0,1)

where, $\hat p$ = sample proportion of inaccurate orders = $\frac{40}{307}$ = 0.13

           n = sample of orders = 307

So, the test statistics =  $\frac{0.13-0.10}{\sqrt{\frac{0.10(1-0.10)}{307} } }$  

                                    =  1.75

The value of z-test statistics is 1.75.

Also, the P-value of the test statistics is given by;

            P-value = P(Z > 1.75) = 1 - P(Z $\leq$ 1.75)

                          = 1 - 0.95994 = 0.04006

Now, at 0.05 level of significance, the z table gives a critical value of 1.645  for the right-tailed test.

Since the value of our test statistics is more than the critical value of z as 1.75 > 1.645, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the rate of inaccurate orders is greater than ​10%.