1. The z test statistic & two-sided tests Section 10.2.3

2. Starter 10.2.3 Write a definition of  as used in hypothesis tests

3. Today’s Objectives Calculate the z test statistic for use in a hypothesis test Use the z test statistic in a two-sided test Use a confidence interval to perform a two-sided test California Standards 17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error. 18.0 Students determine the P- value for a statistic for a simple random sample from a normal distribution.

4. The z test statistic Recall that observations of a variable which varies normally can be standardized into a z-score by the formula If the observation used in this formula is a sample mean (  ), then the result is called the z test statistic and the formula becomes

5. P-value and the z test statistic Remember that P-value is the probability of getting results as extreme as (or more extreme than) those actually achieved if H o is true So the P-value is simply the area to the right (or left) of the z test statistic under the standard normal curve –Area to the right if H a is µ > µ o (where µ o is the assumed mean) –Area to the left if H a is µ < µ o

6. The two-sided Hypothesis Test When H a is that µ is different from some specific value, the P-value has to include areas both right and left of the observation –That’s because the question we ask is how likely are results this far (or farther) from the assumed mean, not results this much greater than or this much less than the assumed mean So P is the total area in the picture is 2 x.045 =.090

7. Example 10.11 The mean blood pressure for American males 35 to 44 years of age is 128 with standard deviation of 15. Is there evidence at the 5% significance level that a certain company’s executives’ blood pressure differs from the national average? Write hypotheses and significance level. –Ho: µ = 128 –Ha: µ  128 –  =.05 (an arbitrary choice, but not unreasonable) A sample of 72 executives at the company has a mean blood pressure of 126.07. Calculate the z-test statistic and P-value and write a conclusion.

8. Answer Calculate the z test statistic. z = (126.07 – 128) / (15 / √72) = -1.09 Find the P-value. P = 2 x normalcdf(-999, -1.09) = 2 x.138 =.276 Note: 2 x normalcdf(0, 126.07, 128, 1.768) is not recommended. The AP test readers want to see the test statistic AND the P-value calculated. Write a conclusion in proper form. Because P > , there is not sufficient evidence to support the claim that the executives’ mean blood pressure differs from the national average.

9. Confidence intervals and two-sided tests Suppose we believe that the true mean of a certain population is µ o. Someone says we are wrong – that the mean is different from µ o. –Note that he does not have to say whether it is too great or too small, just that it is different from µ o. We take a sample, find , and calculate a level C confidence interval. It turns out that the interval does not contain µ o. So is he right? –Probably yes, because if the mean really is µ o then C% of all intervals would contain µ o and this one did not!

10. How does that compare to a hypothesis test? Hypotheses & significance level –H o : µ = µ o H a : µ  µ o Let  = 1 – C If it is true that µ = µ o, then in C% of all samples, we will fail to reject H o. This is just like the statement we made about confidence intervals: –If the mean really is µ o then C% of all intervals would contain µ o. So a level C confidence interval can give exactly the same result as a two-sided hypothesis test where  = 1 – C.

11. Revisit the blood pressure example Use the facts from the previous example and the Zinterval screen on the TI to form a 95% confidence interval. –n = 72  = 15  = 126.07 (Notice that C = 1 -  because we used α =.05) You should find that the C.I. is (122.61, 129.53) Notice that the interval contains the assumed mean of 128 So at the 95% confidence level there is not sufficient evidence to reject H o That is the same result we got in the hypothesis test –Caution: This approach only works for two-sided tests (where H a is µ  µ o ) because confidence intervals are essentially two-sided in that they are built symmetrically around 

12. Homework Read pages 544 – 549 Read pages 554 – 555 Do problems 39, 41 – 44