8-3 Testing a Claim About a Mean 411 Testing a Claim About M When S Is Known In reality, it is very rare to test a claim about an unknown population mean while the population standard deviation is somehow known. For this case of known s, the procedure is essentially the same as earlier in this section, the requirements are the same, but the test statistic, P-value, and critical values are as shown below. P-Value: The P-value is 0.0000 or 0+ (or “less than 0.01” if using Table A-3). Critical Values: The critical values are {1.983 (or {1.984 if using Table A-3). Confidence Interval: The 95% confidence interval is 98.08°F 6 m 6 98.32°F, which does not contain 98.6°F. Step 7: All three approaches lead to the same conclusion: Reject H0. ■ P-Value: The P-value of 0.0000 is less than the significance level of a = 0.05. ■ Critical Values: The test statistic t = -6.64 falls in the critical region bounded by {1.983. ■ Confidence Interval: The claimed mean of 98.6°F does not fall within the confidence interval of 98.08°F 6 m 6 98.32°F. YOUR TURN. Do Exercise 23 “Cell Phone Radiation.” INTERPRETATION Step 8: There is sufficient evidence to warrant rejection of the common belief that the population mean is 98.6°F. CAUTION In days of yore preceding the widespread availability of good technology, this “s known” case was commonly used with this practice: If n 7 30, treat the sample standard deviation s as if it were the known value of s. Because we now have such great technology, we can use the t distribution instead of the normal distribution. Consequently, this “s known” approach is now effectively obsolete. Testing a Claim About a Mean (When s Is Known) Requirements (1) The sample is a simple random sample, (2) the population standard deviation s is known, and (3) either or both of these conditions is satisfied: the sample is large 1n 7 302 or the sample is from a normally distributed population. Test Statistic z = x - mx s2 n P-value: Provided by technology, or use the standard normal distribution (Table A-2) with the procedure in Figure 8-3 on page 380. Critical values: Use the standard normal distribution (Table A-2). KEY ELEMENTS Go Figure Americans spend 37 billion hours in line each year.
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