First, we need to set up our null and alternative hypotheses. The null hypothesis (H0) is that the true proportion of union members who will support a strike is 75% (0.75). The alternative hypothesis (H1) is that the true proportion is less than 75%.

Next, we calculate the sample proportion (p̂) which is 87/125 = 0.696.

Now, we need to perform a hypothesis test to determine if we should reject the null hypothesis in favor of the alternative hypothesis. We can use a one-sample z-test for proportions.

The test statistic (z) is calculated as follows:

z = (p̂ - p0) / sqrt[(p0 * (1 - p0)) / n]

where p0 is the proportion under the null hypothesis, n is the sample size, and sqrt is the square root function.

Substituting the given values:

z = (0.696 - 0.75) / sqrt[(0.75 * (1 - 0.75)) / 125] = -1.44

We compare this test statistic to a critical value from the standard normal distribution, typically -1.645 for a one-tailed test at the 5% significance level. Since -1.44 -1.645, we do not reject the null hypothesis. This means that the company negotiator does not have enough evidence to support his claim that the true percentage is lower than 75%.

If we made a mistake by not rejecting the null hypothesis when it is false, we have committed a Type II error. The consequence of this error is that the company might underestimate the support for a strike among union members, potentially leading to a strike that they were not adequately prepared for.

If we rejected the null hypothesis when it is true, we would have committed a Type I error. The consequence of this error is that the company might overestimate the support for a strike, potentially leading to unnecessary concessions to the union.

Question

First, we need to set up our null and alternative hypotheses. The null hypothesis (H0) is that the true proportion of union members who will support a strike is 75% (0.75). The alternative hypothesis (H1) is that the true proportion is less than 75%.

Next, we calculate the sample proportion (p̂) which is 87/125 = 0.696.

Now, we need to perform a hypothesis test to determine if we should reject the null hypothesis in favor of the alternative hypothesis. We can use a one-sample z-test for proportions.

The test statistic (z) is calculated as follows:

z = (p̂ - p0) / sqrt[(p0 * (1 - p0)) / n]

where p0 is the proportion under the null hypothesis, n is the sample size, and sqrt is the square root function.

Substituting the given values:

z = (0.696 - 0.75) / sqrt[(0.75 * (1 - 0.75)) / 125] = -1.44

We compare this test statistic to a critical value from the standard normal distribution, typically -1.645 for a one-tailed test at the 5% significance level. Since -1.44 > -1.645, we do not reject the null hypothesis. This means that the company negotiator does not have enough evidence to support his claim that the true percentage is lower than 75%.

If we made a mistake by not rejecting the null hypothesis when it is false, we have committed a Type II error. The consequence of this error is that the company might underestimate the support for a strike among union members, potentially leading to a strike that they were not adequately prepared for.

If we rejected the null hypothesis when it is true, we would have committed a Type I error. The consequence of this error is that the company might overestimate the support for a strike, potentially leading to unnecessary concessions to the union.

Knowee AI · Accepted Answer