Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 355 The display presents the confidence interval (63.81, 70.79), along with the values of ̄x and n. Note that the confidence level (98%) is not given. CAUTION Confidence intervals constructed using technology may differ from those constructed by hand due to rounding. The differences are never large enough to matter. Following is MINITAB output for the same example. ������ �������������� ���������� = ������������ ���������������� �� ���������� �������� ���������� ���� �������� ����~ ���� ������ ���� �������� ���� �������� �� ���������� .���� ��������, ���� �������� / The output is fairly straightforward. Going from left to right, ‘‘N’’ represents the sample size, ‘‘Mean’’ is the sample mean ̄x, ‘‘StDev’’ is the population standard deviation ��, and ‘‘SE Mean’’ is the standard error ��∕ √ n. The lower and upper confidence limits of the 98% confidence interval are given on the right. Note that neither the critical value nor the margin of error is given explicitly in the output. Finally, we present EXCEL output for this example. The EXCEL function CONFIDENCE.NORM returns the margin of error. The inputs are the value of ��, the population standard deviation ��, and the sample size n. Step-by-step instructions for constructing confidence intervals with technology are given in the Using Technology section on page 361. Check Your Understanding 9. To estimate the accuracy of a laboratory scale, a weight known to have a mass of 100 grams is weighed 32 times. The reading of the scale is recorded each time. The following MINITAB output presents a 95% confidence interval for the mean reading of the scale. ������ �������������� ���������� = �� �������� ���������������� ���������� �������������� �������� �� ���� ������ �������� ���������� �� �������� ���� �������� �� ���������� ����~ ���� .������ ��������, ������ �������� / A scientist claims that the mean reading �� is actually 100 grams. Is it likely that this claim is true? No 10. Using the output in Exercise 9: a. Find the critical value z��∕2 for a 99% confidence interval. 2.576 b. Use the critical value along with the information in the output to construct a 99% confidence interval for the mean reading of the scale. (101.2143, 103.4911) Answers are on page 366. Objective 3 Describe the relationship between the confidence level and the margin of error More Confidence Means a Bigger Margin of Error Other things being equal, it is better to have more confidence than less. We would also rather have a smaller margin of error than a larger one. However, when it comes to confidence intervals, there is a trade-off. If we increase the level of confidence, we must increase the critical value, which in turn increases the margin of error. Examples 8.5 and 8.6 help explain this idea. EXAMPLE 8.5 Construct a confidence interval A machine that fills cereal boxes is supposed to put 20 ounces of cereal in each box. A simple random sample of 6 boxes is found to contain a sample mean of 20.25 ounces of cereal. It is known from past experience that the fill weights are normally distributed with a standard deviation of 0.2 ounce. Construct a 90% confidence interval for the mean fill weight.
navidi_monk_elementary_statistics_2e_ch7-9
To see the actual publication please follow the link above