Which of the following statement is true?
In testing the hypothesis for the equality of two population proportions, the acceptance region for the null hypothesis will be wider than the confidence interval for two population proportions, assuming the same value of alpha.
A 95% confidence interval estimate for the difference between two population proportions is always symmetrical with zero at the center of the interval.
The test of hypothesis for the equality of two population means based on independent samples with 12 observations from each population will have 22 degrees of freedom.
A quality control manager claims that the proportion of defective units produced by machine I (Population 1) is less than the proportion produced by machine II (Population 2). The correct null hypothesis is Ho: p1 ¨C p2 >= 0.
To determine which statement is true, let's examine each statement:
Statement 1: In testing the hypothesis for the equality of two population proportions, the acceptance region for the null hypothesis will be wider than the confidence interval for two population proportions, assuming the same value of alpha.
To test the hypothesis for the equality of two population proportions, we typically use a hypothesis test like the z-test or the chi-square test. In hypothesis testing, we compare the test statistic (usually calculated using the sample data) to a critical value in order to make a decision about the null hypothesis. The acceptance region for the null hypothesis is the range of values for the test statistic that leads to accepting the null hypothesis.
On the other hand, a confidence interval is an estimate of the range of values within which the true population parameter is likely to fall. In the case of two population proportions, a confidence interval is calculated to estimate the difference between the two proportions.
Comparing the acceptance region for the null hypothesis to the confidence interval, it is not accurate to say that the acceptance region will always be wider than the confidence interval for two population proportions. Their widths depend on factors such as the sample size, the level of significance (alpha), and the variability of the data. Therefore, this statement is not necessarily true.
Statement 2: A 95% confidence interval estimate for the difference between two population proportions is always symmetrical with zero at the center of the interval.
A confidence interval is calculated using sample data and provides an estimated range within which the true population parameter is likely to lie. In the case of comparing two population proportions, the confidence interval estimates the difference between the two proportions.
For a confidence interval to be symmetrical, certain assumptions need to be met, including the data being normally distributed and the sample sizes being large enough. However, the symmetry property does not always hold true for all confidence intervals, particularly when the sample sizes are small or the data are not normally distributed.
Therefore, this statement is not always true.
Statement 3: The test of hypothesis for the equality of two population means based on independent samples with 12 observations from each population will have 22 degrees of freedom.
To test the hypothesis for the equality of two population means, we typically use a t-test when the sample sizes are small or the population variances are unknown. The degrees of freedom in a t-test represent the number of independent observations available for estimating the population mean.
In this case, when we have independent samples with 12 observations from each population, the degrees of freedom for the t-test calculation would be determined by the smaller of the two sample sizes minus 1. Therefore, the correct degrees of freedom in this scenario would be 11, not 22. Thus, this statement is not true.
Statement 4: A quality control manager claims that the proportion of defective units produced by machine I (Population 1) is less than the proportion produced by machine II (Population 2). The correct null hypothesis is Ho: p1 ¨C p2 >= 0.
When comparing two population proportions, the null hypothesis (Ho) usually assumes that there is no difference or no effect between the two proportions. In this case, the null hypothesis should state that the proportion of defective units produced by machine I is greater than or equal to the proportion produced by machine II.
Therefore, the correct null hypothesis would be: Ho: p1 - p2 >= 0.
In conclusion, the only statement that is true is statement 4.