Significance Test

1. Significance Test A claim is made. Is the claim true? Is the claim false?

2. A test of signicance assesses the evidence found in the data against a null hypothesis • in favor of an alternative hypothesis .

3. Null Hypothesis • The null hypothesis H0 is always of the form: • H0 : The null hypothesis says that "the claim is true and that what we observe in the data happened by chance."

4. Alternative Hypothesis • The alternative hypothesis Ha always takes one of three forms: • Two-sided Ha : • One-sided on the high side Ha : • One-sided on the low side Ha : • Note 1: The form of the alternative hypothesis is problem dependent. • Note 2: The alternative hypothesis says "the claim is false and what we observe happened because the claim is false."

5. z Test for a Population Mean • Step 1: Write down the Null and Alternative Hypotheses: • Null Hypothesis Alternative Hypothesis

6. Step 2 Compute the test statistic (z-score)

7. Step 3 • Compute the p- value • run a normalcdf • For a 2 sided test you will need to double this value • The P-value is defined this way: • the probability, computed assuming the null is true, that the test statistic would take a value as extreme or more extreme than that actually observed is called the P-value of the test. • The smaller the P-value, the stronger the evidence against the null hypothesis

8. Step 4 Compare p-value against the alpha level -alpha level is a value that is too extreme to assume that the sample happened just by chance -if p-value falls below the alpha level than the null is rejected -if p-value is above the alpha level than we fail to reject the null • If the P-value is as small or smaller than alpha , we say that the data are statistically significant at that level.

9. Radon is a colorless, odorless gas that is naturally released by rocks and soils and may concentrate in tightly closed houses. Because Radon is slightly radioactive, there is some concern that it may be a health hazard. Radon detectors are sold to home owners worried about this risk. The detectors may be inaccurate. You placed 12 detectors in a chamber where they were exposed to 105 picocuries per liter (pCi/l) of Radon over three days. Here are the readings given by the detectors: 91.9 97.8 111.4 122.3 105.4 95.0 103.8 99.6 119.3 104.8 101.7 96.6 Assume (unrealistically) that you know the standard deviation of readings for all detectors of this type is = 9 and that the population of readings is approximately normal.

10. (i) Give a 95% confidence interval for the mean reading of this type of detector.

11. (ii) Is there significant evidence at the 5% level that the mean reading differs from the true value of 105. State the null and alternative hypotheses and conduct the significance test. Assume the 12 detectors are an SRS. Already stated that normally distributed. Ok to use normal calculations.

12. Conclusion With a P-value as high as 0.74 are sample has very good chance of occurring if the true mean is 105. Therefore, we do not have enough evidence to reject the null and assume the mean reading could be around 105. • In other words: we have no reason to think that the average value of the readings of all such detectors is not 105 and that what we observed, namely that x = 104.13, did not happen by chance

13. Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively. Our interest is in the scores of 55 students in a particular school who received a mean score of 96. We can ask whether this mean score is significantly lower than the regional mean — that is, are the students in this school comparable to a simple random sample of 55 students from the region as a whole, or are their scores surprisingly low?

14. Homework 10.38-40