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Chapter 9: Problem 608

In testing a hypothesis concerned with the value of a population mean, first the level of significance to be used in the test is specified and then the regions of acceptance and rejection for evaluating the obtained sample mean are determined. If the 1 percent level of significance is used, indicate the percentages of sample means in each of the areas of the normal curve, assuming that the population hypothesis is correct, and the test is two-tailed.

Short Answer

Expert verified
Using a two-tailed hypothesis test with a 1 percent level of significance, we find that 1% of sample means fall into the rejection region and 99% of the sample means fall into the acceptance region, assuming that the null hypothesis is true. This is determined by dividing the significance level by 2, finding the corresponding z-scores for both tails (-2.576 and 2.576), and calculating the percentages of sample means in each region.

Step by step solution

01

Identify the significance level and tails of the test

The given level of significance is 1 percent, which is written as 0.01. Since it's a two-tailed test, we will divide this significance level by 2 to find the percentage of sample means in each of the tails of the normal curve.
02

Determine the z-scores corresponding to the significance level

In a two-tailed test, we divide the significance level by 2 so that we can consider both the left and right tails of the distribution. Thus, we have 0.01 / 2 = 0.005 for each tail. Using the z-table or a calculator, find the z-scores corresponding to the 0.005 in the left tail and 1 - 0.005 (0.995) in the right tail. We get the z-scores approximately as -2.576 and 2.576, respectively.
03

Calculate the percentages of sample means that fall into the acceptance and rejection regions

Now that we have the z-scores for both tails, we can determine the percentage of sample means in each region: 1. Rejection region (two tails): The total percentage in the rejection region is the sum of the percentages in both tails, which is 0.005 + 0.005 = 0.01, or 1%. 2. Acceptance region (middle): To find the percentage of sample means in the acceptance region, we subtract the total percentage in the rejection region from 100%. So, 100% - 1% = 99%. In summary, based on a two-tailed hypothesis test with a 1 percent level of significance, we can conclude that 1% of sample means fall into the rejection region and 99% of the sample means fall into the acceptance region, assuming that the null hypothesis is true.

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