Refer to Exercise 7.11. Suppose that in the forest fertilization problem the population standard deviation of basal areas is not known and must be estimated from the sample. If a random sample of n = 9 basal areas is to be measured, find two statistics g1 and g2 such that
7.11 A forester studying the effects of fertilization on certain pine forests in the Southeast is interested in estimating the average basal area of pine trees. In studying basal areas of similar trees for many years, he has discovered that these measurements (in square inches) are normally distributed with standard deviation approximately 4 square inches. If the forester samples n = 9 trees, find the probability that the sample mean will be within 2 square inches of the population mean.
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Chapter 7 Solutions
Mathematical Statistics with Applications
- An engineer is comparing voltages for two types of batteries (K and Q) using a sample of 7070 type K batteries and a sample of 8585 type Q batteries. The type K batteries have a mean voltage of 8.848.84, and the population standard deviation is known to be 0.3030.303. The type Q batteries have a mean voltage of 9.059.05, and the population standard deviation is known to be 0.3670.367. Conduct a hypothesis test for the conjecture that the mean voltage for these two types of batteries is different. Let μ1μ1 be the true mean voltage for type K batteries and μ2μ2 be the true mean voltage for type Q batteries. Use a 0.010.01 level of significance. Step 3 of 4 : Determine the decision rule for rejecting the null hypothesis H0H0. Round the numerical portion of your answer to three decimal places.arrow_forwardListed in the data table are amounts of strontium-90 (in millibecquerels, or mBq, per gram of calcium) in a simple random sample of baby teeth obtained from residents in two cities. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. City #1 City #2102 11786 78121 100113 85101 85104 107213 110100 111290 130100 133282 101145 209 Construct a confidence interval suitable for testing the claim that the mean amount of strontium-90 from city #1 residents is greater than the mean amount from city #2 residents. _____mBq<μ1−μ2<____mBqarrow_forwardListed in the data table are amounts of strontium-90 (in millibecquerels, ormBq, per gram of calcium) in a simple random sample of baby teeth obtained from residents in two cities. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. City_#1 City_#2 100 117 86 61 121 100 119 85 101 89 104 107 213 110 116 111 290 142 100 133 283 101 145 209 The test statistic is The P-value is construct a confidence interval suitable for testing the claim that the mean amount of strontium-90 from city #1 residents is greater than the mean amount from city #2 residents. ____mBq<μ1−μ2<____mBqarrow_forward
- Recently, the annual number of driver deaths per 100,000 for the selected age groups was as follows: Age Number of Driver Deaths per 100,000 16–19 38 20–24 36 25–34 24 35–54 20 55–74 18 75+ 28 Use the 4 steps of hypothesis testing to see if the prediction is significant with a criteria of alpha=.05 on the following data For each age group, pick the midpoint of the interval for the X value. (For the 75+ group, use 80.)arrow_forwardA dietitian wishes to see if a person’s cholesterol level will be changed if the diet is supplemented by a certain mineral. Four subjects were pre-tested, and they took the mineral supplement for a 6-week period. The results are shown in the table. Is there sufficient evidence to conclude that the population mean of cholesterol levels has been changed after six weeks at α=0.2α=0.2? Assume that the differences are from an approximately normally distributed population. Subject Cholestrol Level (mg/dl) Cholestrol Level after 6 Weeks (mg/dl) dd ¯dd¯ (d−¯d)2(d-d¯)2 1 206 217 11 2 219 184 -35 3 202 204 2 4 213 205 -8 Total -30 a) Calculate the mean, the sum of the squared deviation from the mean, and the standard deviation of differences. Do not include the unit for each answer: ¯d=d¯= (do not round) ∑(d−¯d)2=∑(d-d¯)2= (do not round) sd=sd= (rounded to one decimal place) b) Perform the hypothesis test in the following steps: Step 1.…arrow_forwardListed in the data table are amounts of strontium-90 (in millibecquerels, or mBq, per gram of calcium) in a simple random sample of baby teeth obtained from residents in two cities. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. City_#1 City_#2 100 117 86 84 121 100 120 85 101 90 104 107 213 110 136 111 290 126 100 133 289 101 145 209 Use a 0.01 significance level to test the claim that the mean amount of strontium-90 from city #1 residents is greater than the mean amount from city #2 residents. The test statistic is The P-value State the conclusion for the test. a ) reject, There is sufficient evidence to support the claim that the mean amount of strontium-90 from city #1 residents is greater. b )reject, There is not sufficient evidence to support the claim that the mean amount of strontium-90 from city #1 residents is greater. c ) fail to reject,…arrow_forward
- A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.33. A dietician is asked to test this claim and finds that a random sample of 24 servings has a variance of 1.37. At α=0.01, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed. Complete parts (a) through (e) below. (a) Write the claim mathematically and identify H0 and Ha. A. H0: σ2≤1.33 (Claim) Ha: σ2>1.33 B. H0: σ2≠1.33 Ha: σ2=1.33 (Claim) C. H0: σ2≥1.33 Ha: σ2<1.33 (Claim) D. H0: σ2=1.33 (Claim) Ha: σ2≠1.33 (b) Find the critical value(s) and identify the rejection region(s). The critical value(s) is(are) enter your response here. (Round to two decimal places as needed. Use a comma to separate answers as needed.) Choose the correct statement below and fill in the corresponding answer boxes. A. The…arrow_forwardAssume that you have a sample of n1=7 , with the sample mean X1=44 , and a sample standard deviation of S1=6 , and you have an independent sample of n2=6 from another population with a sample mean of X2=32 and the sample standard deviation S2=5 . Assuming the population variances are equal, at the 0.01 level of significance, is there evidence that μ1>μ2?arrow_forwardThe desired percentage of SiO2 in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained samples are analyzed. Suppose that the percentage of SiO2 in a sample is normally distributed with ? = 0.32 and that x = 5.21. (Use ? = 0.05.) (a) Does this indicate conclusively that the true average percentage differs from 5.5?State the appropriate null and alternative hypotheses. H0: ? = 5.5Ha: ? ≠ 5.5H0: ? = 5.5Ha: ? ≥ 5.5 H0: ? = 5.5Ha: ? < 5.5H0: ? = 5.5Ha: ? > 5.5 Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. Do not reject the null hypothesis. There is sufficient evidence to conclude that the true average percentage differs from the desired percentage.Reject the null hypothesis. There is sufficient evidence…arrow_forward
- Refer to Exercise 8.S.1. Compare the before and after populations using a t test at α = 0.05. Use a nondirectional alternative.arrow_forwardDr. Romanoff reported the following in a journal: “F (5, 106) = 10.09, p = .04.” Should Dr. Romanoff state that there are significant differences among the variable means at a .05 alpha level?arrow_forward25. The State of California claims the population average of the amount of ice cream each Californian eats in the month of September is 6.85 pints with population standard deviation of 1.35 pints. An SRS of 500 Californians resulted in a sample average of 6.75 pints eaten per person in the month of September . At alpha=0.05, is there evidence to support the State of California's claim that Californians eat an average of 6.85 pints of ice cream in the month of September? Write a conclusion using the context of the problem.arrow_forward
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