1. Warm Up Researches tested 150 farm raised salmon for organic contaminants. They found the mean of the carcinogenic insecticide mirex to be.0913 parts per million, with standard deviation.0495 ppm. As safety recommendation to recreational fishers, the EPA recommended screening value for mirex is.08 ppm. Are farmed salmon contaminated beyond the level permitted by the EPA?

2. Warm Up Continue Find a 95% confidence interval

3. Ch. 23 – Inferences about Means Day 3: Normality Condition Part VI – Learning about the World

4. The t-distribution is “robust” The t-distribution is more “robust” than the z-distribution This means that the results of tests and confidence intervals are not strongly influenced when the conditions of the procedure are violated

5. Can we use t no matter what? For smaller samples, we still need to have an approximately normal distribution For larger samples, we can proceed even if the population distribution is not normal Since we don’t know the shape of the population distribution, we will be looking at the shape of the sample data to tell us whether we can use t We can only do this if we have the actual data to look at

6. Guidelines For t-procedures, we will be more “relaxed” about the normality condition, but how relaxed we are will depend on the sample size n < 15: Use t if the sample data are close to normal (approximately symmetric & unimodal) 15 ≤ n < 40 : Use t except when the sample data are clearly skewed or have outliers n ≥ 40: Use t even for clearly skewed sample data, but use caution with outliers

7. So how do we check this condition? So far, all of the problems we have done have given us only the summary statistics, not the actual data When this happens: n < 15 15 ≤ n < 40 n ≥ 40 ConditionCheck Pop approx. normal or large n Assume the sample data is approx. symmetric with no outliers ConditionCheck Pop approx. normal or large n Assume the sample data has no outliers or strong skew ConditionCheck Pop approx. normal or large n n ≥ 40, so t can be used

8. So how do we check this condition? Sometimes the problem will give you actual data from the sample In this case, the conditions are the same, but you must draw a quick plot (boxplot or histogram) to verify your assumption If the sample data has no strong skew or outliers, we will assume that the population data is approximately normal

9. Example An anthropologist wants to know if the average height of the men of a certain tribe is different from that of the rest of the residents of that region. The average height for the region is 67 inches. He takes a sample of 20 tribe members. The results are as follows. What can he conclude? 62 70 64 61 69 66 62 64 65 68 60 62 63 64 63 65 64 65 70 65

10. ConditionCheck Random Sample n < 10%N Pop approx. normal or large n Since p < α, reject H o. There is enough evidence to conclude that the average height of men in this tribe is different from the rest of the region. μ = the mean height of men in the tribe H 0 : μ = 67 in. H a : μ ≠ 67 in. One-sample t-test, α =.05, df = 19 Assume Assume > 200 tribe members (n=20) Data has no outliers or strong skew:

11. Homework 23-3 P. 554 #22, 34, 35 For # 22 –You take a random sample of 40 hot dogs and find the mean sodium content to be 310 with a standard deviation of 36. Construct a 95% confidence interval. For # 34 (b) –Requires a t test