Sampling Distributions, Confidence Intervals and Hypothesis Tests for a Mean

Transcription

1 1. A boe health study looked at the daily itake of calcium (mg) for 38 wome. I the populatio of wome, the stadard deviatio is kow to be mg. However, the researchers are cocered that the mea calcium itake for this populatio is ot meetig the RDA level of 1200 mg, that is, the populatio mea is less tha the 1200 mg level. They wish to test this theory usig a 5% sigificace level. a. State the hypotheses. Ho: μ = 1200 versus Ha: μ < 1200 b. The researchers fid that the mea for the sample is mg. Calculate the test statistic ad p value based this data. x z p value = P(Z < 3.41) = (from Table A.3) Test statistic = 3.41 p value = c. Circle the appropriate decisio: Reject Ho Fail to reject Ho Explai your choice. Because the p value < 0.05 d. Use your decisio to complete the real world coclusio that ca be made based o this test: There (circle oe) is is ot sufficiet evidece to say the average calcium itake for wome is below the RDA level of 1200 mg. e. Sketch a picture of the p value i terms of a area uder a distributio. The p value is the blue shaded area i the picture at the left. What the p value tells: If the true mea were 1200, the probability that a sample of 38 wome would produce a mea as low as is , or 0.03%. So the size of the blue shaded area is 0.03%. Page 1 of 5

2 2. Compay records show that drivers get a average of 32,500 miles with a stadard deviatio of 1,444 miles o a set of All Weather tires before a uacceptable level of wear o tires results. Hopig to improve the average, the compay has added a ew polymer to the rubber that should help protect the tires from wear caused by extreme weather. The results of a prelimiary study from twety radomly selected drivers who tested the ew tires gave a sample mea of 33,449 miles. Coduct the appropriate test usig a 1% sigificace level. a. Clearly state the hypotheses to be tested. H0 : 32500, H a : b. Coduct the test of the hypotheses specified i part (a). x z p value = P(Z > 2.94) = (from Table A.3) Test statistic = 2.94 p value = Coclusio (circle oe): Reject Ho Fail to reject Ho 3. Suppose we wish to test the hypotheses H 0 : µ = 10 vs. H a : µ < 10, where µ represets the mea age of o high school aged childre who are members of a large gymastics club i a metropolita area. A hypothesis test was coducted which resulted i a test statistic of 2.2 ad a p value of What does the p value tell us? (Circle oe) a) The probability that the ull hypothesis is true. b) The probability that the alterative hypothesis is true. c) The largest level of sigificace at which the ull hypothesis ca be rejected. d) The smallest level of sigificace at which the ull hypothesis ca be rejected. 4. It s a good year for MBA grads was the title of a article i the A Arbor News, (5/30/05). Oe of the parameters of iterest was the populatio mea expected salary, μ (i dollars). A radom sample of 1000 studets who fiished their MBA this year (from 129 busiess schools) resulted i a 95% cofidece iterval for μ of (83700, 84800). a. What is the value of the sample mea? (Iclude your uits.) sample mea = midpoit of iterval = ( )/2 = 84250, or $84,250 b. The expected average earigs for such graduates i 2004 was $76,100. Suppose we wish to test the followig hypotheses at the 5% sigificace level: Ho: μ = versus Ha: μ Our decisio would be (circle oe): Reject Ho Fail to reject Ho Because The value of is ot i the 95% cofidece iterval. Page 2 of 5

3 5. Trash Bag Stregth: Jay Krug works for a distributio compay that purchases trash bags from a vedor. The bags are required to have a breakig stregth of at least 47 pouds. The bag vedor claims that its productio process is relatively stable ad produces bags with a mea breakig stregth of 53 pouds ad a variace of 17.6 pouds. A week ago Jay cosulted with you askig how he should go about checkig this claim. You suggested obtaiig some data. So Jay strikes a agreemet with the vedor that permits him to sample from the vedor s productio process. A sample of 49 bags was radomly selected ad the mea breakig stregth for the sample was foud to be 51 pouds. Jay does remember that sample meas are statistics which vary from sample to sample, but he is ot sure if his result reflects more variatio tha due to chace. He seeks your help i iterpretig his result. a. Suppose the bag vedor s claim is true ad may radom samples of 49 bags were obtaied ad for each sample, the sample mea was computed. What is the distributio (model) for the possible values for the sample mea? (Be specific.) The sample mea will have a distributio that is approximately ormal with a 2 mea of 53 pouds ad a variace of pouds. OR You ca just put: N(53,17.6/49) => N(53,0.36) [Note: we ca apply the CLT here sice >25, so it does t matter if the breakig stregths follow a ormal distributio or ot.] b. Assumig the bag vedor s claim is true, what is the probability of obtaiig a sample mea as far or eve farther below the process mea tha that observed by Jay? Show all work PX ( 51) P Z PZ ( 3.33) Fial aswer: c. What does your aswer i part (c) suggest about the validity of the bag vedor s claim? (circle oe) The vedor s claim is reasoable. The vedor s claim is ot reasoable. d. Suppose the sample size had bee larger. How would the distributio i part (a) have chaged? (circle all that apply) the mea would have icreased the mea would have decreased the variability would have icreased the variability would have decreased [Note: This example illustrates the coectio betwee the samplig distributio for the mea ad hypothesis testig.] Page 3 of 5

4 6. A maufacturer of light bulbs selects at radom 100 light bulbs from their productio lie ad measures the lifetime of each light bulb i hours (that is, how log the light bulb stay o before it burs out). The resultig data is show i the histogram below. The maufacturer would like to use this data to estimate the populatio mea lifetime of its light bulbs. a. Based o the histogram, do the data appear to come from a ormal populatio? If yes, state your estimate for the mea of the populatio. If o, state the shape of the populatio. No the distributio does ot appear ormal but rather skewed to the right. b. The Cetral Limit Theorem helps us i estimatig the populatio mea lifetime of the light bulbs because: (circle oe) It tells us that the test statistic will have a distributio that has the same shape as the populatio. It tells us that the distributio of the sample mea will have the same shape ad mea as the populatio distributio, but that the variace will be equal to the populatio variace divided by 100. It tells us that the distributio of the sample mea will be approximately ormal with its mea equal to the populatio mea ad its variace equal to the populatio variace divided by 100. Page 4 of 5

5 7. Recetly the U.S. 5 cet coi (ickel) has bee redesiged. A egieer for a vedig machie would like to estimate the average weight of this ew coi. He radomly selects 41 of the redesiged ickels. From this sample he foud that the average weight was 4.86 grams. Assume that the populatio stadard deviatio of weights is 0.5 grams. a. Fid a 90% cofidece iterval for the populatio mea weight of the ew ickel. 0.5 x z / ±0.128 Aswer: ( to ) b. Fid a 90% predictio iterval for the weight of a ew ickel. 1 1 x z (1.645)(0.5) 1 / ±0.832 Aswer: ( to ) 8. Normal huma body temperature is widely believed to be 98.6 F with a stadard deviatio of 0.68 F. A premed studet would like to see if this average is accurate. He radomly samples 52 people ad fids their average body temperature to be 98.8 F. Calculate a 95% cofidece iterval for the populatio average body temperature x z / ±0.185 Aswer: ( to ) Page 5 of 5