Question

Testing Claims About Proportions. In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Drug Screening The company Drug Test Success provides a “1-Panel-THC” test for marijuana usage. Among 300 tested subjects, results from 27 subjects were wrong (either a false positive or a false negative). Use a 0.05 significance level to test the claim that less than 10% of the test results are wrong. Does the test appear to be good for most purposes?

Short answer

Null hypothesis: The proportion of results that are wrong is equal to 10%.

Alternative hypothesis:The proportion of results that are wrong is less than 10%.

Test statistic: -0.577

Critical value: -1.645

P-value: 0.2820

The null hypothesis is failed to reject.

There is not enough evidence to support the claim that the proportion of test results that are wrong is less than 10%.

Although the null hypothesis isnot rejected, the proportion of inaccurate results isjust 9%, which can be considered low.

As a result, the test appears to be suitable for a majority of purposes.

Step-by-step solution

Given information

A sample of 300 subjects is tested for marihuana usage. Out of the 300 tested subjects, 27 of the results are wrong.

Hypotheses

The null hypothesis is written as follows.

The proportion of results that arewrong is equal to 10%.

\({H_0}:p = 0.10\).

The alternative hypothesis is written as follows.

The proportion of results that arewrong is less than 10%.

\({H_1}:p < 0.10\).

The test is left-tailed.

Sample size, sample proportion, and population proportion

The sample size is n=300.

The sample proportion of the results is computed below.

\[\begin{array}{c}\hat p = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{results}}\;{\rm{that}}\;{\rm{were}}\;{\rm{wrong}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{results}}\;}}\\ = \frac{{27}}{{300}}\\ = 0.09\end{array}\].

The population proportion of the results that are wrong is equal to 0.10.

Test statistic

The value of the test statistic is computed below.

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.09 - 0.10}}{{\sqrt {\frac{{0.10\left( {1 - 0.10} \right)}}{{300}}} }}\\ = - 0.577\end{array}\).

Thus, z=-0.577.

Critical value and p-value

Referring to the standard normal table, the critical value of z at\(\alpha = 0.05\)for a left-tailed test is equal to -1.645.

Referring to the standard normal table, the p-value for the test statistic value of -0.577 is equal to 0.2820.

As the p-value is greater than 0.05, the decision is fail to reject the null hypothesis.

Conclusion of the test

There is not enough evidence to support the claim that the proportion of results that is wrong is less than 10%.

Although the null hypothesis is failed to reject, the proportion of incorrect results equal to 9% is reasonably low.

Thus, it can be said that the test seems good for most purposes.