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# Elements of Statistics 1 MATH 130

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This 4 page Class Notes was uploaded by Vernie McCullough on Thursday October 15, 2015. The Class Notes belongs to MATH 130 at Millersville University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/223536/math-130-millersville-university-of-pennsylvania in Mathematics (M) at Millersville University of Pennsylvania.

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Date Created: 10/15/15

Millersville University Department of Mathematics MATH 130 Review of Con dence Intervals and Hypothesis Testing April 7 2008 1 E0 00 Five thousand college graduates were asked how many years were required to earn their bachelors degrees The sample mean was 52 years with a standard deviation of 17 years Find the 98 con dence interval for the mean graduation time for all college students This is a large sample n 5000 gt 30 with mean E 52 and s 17 Note that the population standard deviation is not known so we will use the Student s t distribution for the distribution of the sample means At the 98 con dence level the value of oz 002 and thus 042 001 The critical value is tut2 2326 according to Table V Thus the margin of error in this estimate of the mean is s 17 Eta 7 2326 7 2 ltx5000 Note that we are rounding the margin of error to one decimal place just like the mean and standard deviation Therefore the 98 con dence interval estimate of the mean number of years of college before graduation for all college students is 27EEED 270152004505 If the standard deviation in the price of a new car is 147138 how many car prices must be sampled if the error of the sample mean should be no greater than 2000 at the 90 con dence level According to the problem statement 0 14138 The margin of error for the sample should be E 2000 At the 90 con dence level the value of oz 010 and thus 042 005 The critical value is 2042 1645 according to Table V Thus the sample size needed for this estimate is 2 n U2 Notice that we must always round sample sizes up to the nept whole number 1645 2 4444 141382 13522 25136 2000 ln researching automobile maintenance costs 15 service stations were asked the price of an oil change The mean was 1756 with a standard deviation Of 341 Construct the 99 con dence interval for the mean cost of an oil change This is a small sample n 15 S 30 with mean E 1756 and s 341 The population standard deviation is unknown so we will use the Student s t distribution for the distribution of the sample means The number of degrees of freedom for this sample is df n 71 15 71 14 At the 99 con dence level the value ofa 001 4 Thus 042 0005 and tag 2977 according to Table V Thus the margin of error in this estimate of the mean is 2977 m 262 Note that we are rounding the margin of error to two decimal places just like the mean and standard deviation Therefore the 99 con dence interval estimate of the average price of an oil change is E ton2 2 7 E E 1756 i 2621756 262 1494 2018 An engineer claims that a new fuel injection design increases the mean mileage on a car above its current 31 mpg a Express the claim in symbolic form Ifu represents the mean mileage of cars with the new fuel injection design then the claim above can be eacpressed in the symbolic form it gt 31 A C7 V Identify the null hypothesis Since the null hypothesis always contains a condition of equality then the null hypothesis can be eacpressed symbolically as H0 u 31 A O V Identify the alternative hypothesis The alternative hypothesis does not contain a condition of equality thus the alter native hypothesis is the original claim in this case It can be eacpressed symbolically as H1 u gt 31 Identify the test as being left tailed right tailed or two tailed The alternative hypothesis can be used to determine whether a test is left tailed right tailed or two tailed The inequality symbol always points in the direction of the tail of the test In this eccample the inequality symbol is quotgreater than gt and therefore this is a right tailed test A CL A D V Assuming the conclusion is to reject the null hypothesis state the conclusion in nontechnical terms Since the original claim is the alternative hypothesis and the null hypothesis is rejected the conclusion can be stated as quotThe sample data support the claim that the mean mileage of cars with the new fuel injection design is greater than 3 mpg A i h V Assuming the conclusion is failure to reject the null hypothesis state the conclu sion in nontechnical terms Since the original claim is the alternative hypothesis and the null hypothesis is not rejected the conclusion can be stated as 5 03 quotThere is not su cient sample data to support the claim that the mean mileage of cars with the new fuel injection design is greater than 3 mpg In a study of consumer habits researchers designed a questionnaire to identify com pulsive buyers For a sample of consumers who identi ed themselves as compulsive buyers questionnaire scores have a mean of 083 and a standard deviation of 024 Assume that the subjects were randomly selected and that the sample size was 32 At the 001 signi cance level test the claim that the self identi ed compulsive buyer population has a mean greater than 021 the mean for the entire population Does the questionnaire seem to be effective in identifying compulsive buyers In this eccample the important sample statistics are the sample size n 32 the sample mean T 083 and the sample standard deviation 5 024 The symbolic form of the claim made above is u gt 021 where it represents the popula tion mean score of compulsive buyers on the questionnaire Since this claim does not contain a condition of equality it must be the alternative hypothesis H1 u gt 021 Therefore the null hypothesis is H0 u 021 The test to be performed is a right tailed test at the oz 001 signi cance level on a large sample since it gt 30 The population standard deviation is unknown so we will use the Student s t distribution to represent the distribution of the sample means Thus from Table V the critical value of CV ta 2326 The test statistic is T7 u i 083 i 021 s mam35 The test statistic is much larger than the critical value and thus the test statistic lies in the critical region of this right tailed test Since the test statistic falls into the critical region then the null hypothesis is rejected Now since the original claim is the alternative hypothesis and the null hypothesis is rejected the conclusion can be stated as TS to 1461 7 quotThe sample data support the claim at the 001 signi cance level that the score of compulsive buyers on the questionnaire is higher than the score of the entire population The questionnaire seems to be e ective at identifying compulsive buyers According to the Insurance Information Institute the mean expenditure for auto insur ance in the United States was 774 in 2002 An insurance salesman obtains a random sample of 35 auto insurance policies and determines the mean expenditure to be 735 with a standard deviation of 4831 ls there enough evidence to conclude that the mean expenditure for auto insurance is different from the 2002 amount at the oz 001 level of signi cance Here the important sample statistics are the sample size n 35 the sample mean E 735 and the sample standard deviation 5 4831 The symbolic form of the claim made by the insurance salesman is u 74 774 where n represents the population mean eccpenditure for auto insurance This claim is the alternative hypothesis H1 u 74 774 Therefore the null hypothesis is H0 u 774 The test to be performed is a two tailed test at the 04 001 signi cance level on a population whose standard deviation 0 is unknown Thus we will use the Student s t distribution The number of degrees offreedom is df n 7 1 34 Thus from Table V the critical value of CV tag i2728 The test statistic is fin 7 7357774 7 5m 4831 N35 7 Since the test statistic falls into the critical region the null hypothesis is rejected Now since the original claim is the alternative hypothesis and the null hypothesis is rejected the conclusion can be stated as TS to 74776 quotThe sample data support the claim at the 001 signi cance level that the mean eccpenditure for auto insurance is di erent from 774

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