Advanced Math Topics Chapter 10 Review

a) SectionGivensSteps Tests Concerning Means for Large Samples -population or claim average and σ 1) 2) 2-tail or 1-tail? Divide the significance % by 2 or not? 3) Draw a picture with %’s and z-scores 4) Final conclusion -Accept/Reject claim -Is the company paying significantly less? -Are the students significantly less skilled? -sample larger than 30 and a sample average

Lay’s claims that each bag of potato chips weighs 12 ounces on average, with a standard deviation of 0.8 ounces. A consumer’s group tests this claim by weighing 49 randomly selected bags. The sample mean of the selected bags is 11.8 ounces. Should we reject Lay’s claim (is the difference from the claim to the sample significant)? Use a 5% level of significance. μ = 12 Since the alternative hypothesis is that μ ≠ 12, this claim can be shown if the sample mean is much greater OR much less than 12. Thus, it is called a two-tailed test z = 1.96z = If the results of the sample have a z-score greater than 1.96 or less than -1.96, then we can reject Lay’s claim. z = Since the z-score is in the acceptance region, our decision is that must accept the claim of Lay’s that their average bag is 12 ounces. The difference is not significant. The significance level is the probability that the sample mean falls within the rejected region when the null hypothesis is true. Basically, it is the total rejection area for the sample mean. This can be replaced with s, the sample standard deviation.

b) SectionGivensSteps Tests Concerning Means for Small Samples -population or claim average 1) 2) 2-tail or 1-tail? Divide the significance % by 2 or not? 3) Draw a picture with %’s and t-scores using the back of the with df = n – 1 4) Final conclusion -Accept/Reject claim -Is the company paying significantly less? -Are the students significantly less skilled? -sample < 30, a sample average, and a sample standard deviation

A new health pill is being used at a hospital to help fight cholesterol. The manufacturer claims that anyone with high cholesterol who takes this pill will lose 15 mg of cholesterol within a month. A doctor believes this is inflated, & gives this pill to six people with high cholesterol and finds that they lose an average of 12 mg with a standard deviation of 4 mg. Should we reject the manufacturer’s claim? Use a 5% level of significance. μ = 15 Since the sample is small, we use the t-table in the back. The degrees of freedom is = 5. You can either draw a picture and label the rejection region or divide the significance level by 1 (because it is a one-tail test) to find the 0.05/1 = t t = t = Since the t-score is in the acceptance region, we cannot reject the manufacturer’s claim that the average amount of cholesterol lost is 15 mg.

c) SectionGivensSteps Tests Concerning Differences Between Means for Large Samples -two sample sizes > 30 1) 2) 2-tail or 1-tail? Divide the significance % by 2 or not? 3) Draw a picture with %’s and z-scores 4) Is the difference significant, yes or no? -two sample averages -two sample standard deviations

A sociologist claims that a recent female college graduate earns less than a recent male college graduate. A survey of 30 women found their average salary to be $29,000 with a standard deviation of $600. A survey of 40 men found their average salary to be $29,700 with a standard deviation of $900. Do these figures support the claim that women earn less? Use a 1% level of significance. z = z = One tail with 1% significance There is a significant difference between the starting salaries of men and women.

d) SectionGivensSteps Tests Concerning Differences Between Means for Small Samples -same as part c) but two sample sizes < 30 1) 2) 2-tail or 1-tail? Divide the significance % by 2 or not? 3) Draw a picture with %’s and t-scores using df = n + n – 2 4) Is the difference significant, yes or no?

A chemist at a paint factory claims to have developed a new paint that will dry quickly. The manufacturer compares this new paint with its fastest drying paint. He tests 5 cans of each and finds that the current paint has an average drying time of 45.4 minutes with a standard deviation of minutes. The new paint has an average drying time of 43.4 minutes with a standard deviation of 1.14 minutes. Using a 5% level of significance, test the claim of the chemist. Steps 1) Use the formula… 2) Use the formula and the result from step 1… 3) Figure if it’s a 1-tail or 2-tail test and look up the appropriate column in the t-table in the back of the book with df = n 1 + n 2 – 2 4) Compare your answers from steps 2 and 3 to see if you can conclude that there is a significant difference between the two means. Steps 1) = ) = 1.98 Round however you wish, the more the merrier! 3)It is a 1-tail test so look up t 0.05 with df = 8 t = )Since our sample t-value of 1.98 is outside the t-value from the chart, the answer is… “There is a significant difference. The paint does dry quicker!”

e) SectionGivensSteps Tests Concerning Proportions -a % claim 1) 2) 2-tail or 1-tail? Divide the significance % by 2 or not? 3) Draw a picture with %’s and z-scores 4) Accept/Reject? -a sample with two numbers…example 21/55

A senator claims that 55% of the of Americans are in favor of governmental health care. To test this claim, a newspaper editor selected a random sample of 1000 people and 490 of them said that they supported governmental health care. Is the senator’s claim justified? Use a 1% level of significance. p is 0.55 p is 490/1000 =0.49 > z = tail or 1-tail? 2-tail How big is each rejection region? 0.5% How big is each middle region? 49.5% =.495 What are the z-scores in the picture? z = and z = 2.58 Reject the senator’s claim!