Statistics for Business Students
Statistics for Business Students Stat 204
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This 38 page Class Notes was uploaded by Ena Kris on Tuesday September 22, 2015. The Class Notes belongs to Stat 204 at Colorado State University taught by Mark Dahlke in Summer 2015. Since its upload, it has received 19 views. For similar materials see Statistics for Business Students in Statistics at Colorado State University.
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Date Created: 09/22/15
Formulas Stat 204 7 Exam 3 updated Spring 2012 i 7 Zn IQR 7 Q3 7 91 2 ml7E 7 Ez7 82 n7 7 n71 Z Q1 7 175IQR Q3 1 5IQR Eltzgt 7 M 7 Emma WWW 02 Z I 7 WWW E 121 7 2 1 x n71 167 M 7 90111 p lt17pgtlt gt W 7 Z Lg Pz S 10 17 6710 7 a 7 s IiZag rind2W dfn71 pi gum1 90 7 17645 95 71796 99 7 27576 T790 7 T790 7 2 a ifs dfinil F PO W 0392 0392 E17E2iZa2 771 77 Z 7 T17T27D0 a U2 fin 2 2 7 7 7 7 31 2 7 7 1112D0 7 11 7 12 i tut2 n1 n2 df 7 formulaD t 7 5 5 df 7 formulaD i 772 3i i 7 71 ha 7 71 42 7 n 7 sd 7 n 7 7 F1 1 7 7 n7n7 pl 7 02 i 2042 P1ltn1P1 P2ltn2p2 Z 7 P1 P2 Where p 7 1101 2102 2617z agtn711n72 1 Fivesteps of hypothesis testing 1 State the hypotheses 2 Specify a level of signi cangce 3 Compute the test statistic from sample data 4 Find the pValuel 5 Reject H0 if p Value g at ANOVA table for a completely randomized design Source of variation Sum of squares df Mean square F Treatments SSTR k 7 l MSTR Aflng Error Residuals SSE 7LT 7 k MSE Total SST 7LT 7 l i earner 7 my 313 7 n71 i n71 My 7 sisy SLR formulas Ey o ir bo 1111 b 7 Eelawry b Iib T 1 7 riff 2 I27 2 2 0 7 y 1 Production workers example In May 2012 the Bureau of Labor Statistics reported the mean hourly wage for US production workers as 1184 If a sample of 75 production workers the following year had a mean of 1220 per hour do we have evidence that an increase occurred in mean hourly earnings Use 1 005 and assume the population standard deviation is 145 Supermarket checkout example The reported average waiting time in a supermarket checkout line for the city of Baltimore is 472 minutes We decide to test this claim and determine if the actual mean waiting time differs from the reported value Our sample of 120 Baltimore shoppers has a mean waiting time of 522 minutes What is the conclusion at 1 005 if we assume the population variance is 1024 minutes 1 Use the pValue method to conduct this hypothesis test Supermarket checkout page 2 2 Conduct the hypothesis test using the confidence interval method Supermarket checkout page 3 3 Conduct the hypothesis test using the critical value method Milk consumption example In 2008 the U S Department of Agriculture reported the annual per capita milk consumption for adolescents and adults as 139 gallons A small town in the Midwest believed milk consumption was higher there than the national average and wanted to support that opinion A sample of 16 adolescents and adults from that town had a mean annual consumption of 164 gallons with a standard deviation of 48 If the annual consumption amounts for the citizens of the town is approximately normal what is your conclusion at 1 005 Password example USA Today reported that at least 79 of computer users have a unique password for every online account In a carefully constructed survey of 300 computer users we find that 229 of them have a unique password for every online account At 1 005 should we reject USA Today s claim Car mpg example The mean mpg of a certain car model is reported as 25 We think this is incorrect If x represents the mpg for an indiVidual car a random sample of 22 cars of this model reports the following data a 264 and 2xl 902 1371 Assume that x has an approximately normal distribution 1 Test our claim at 1 005 Car mpg page 2 2 Find the 95 confidence interval for u Carefully explain why this confidence interval agrees with the conclusion of our hypothesis test in the previous problem College sports example It was claimed that at U S universities and colleges the average number of male sports offered was larger than the average number of female sports offered A random selection of 95 schools resulted in the following information Gender n J 1 Male 45 86 2 Female 50 79 Assume identical population variances of 1089 and test the claim at 1 005 Missing persons example The missing persons data below is from samples of five random years different years for each sample Population n a s 1 Juvenile 5 63356 2808 2 Adult 5 353 87 2631 1 Find the 95 confidence interval for the difference in average number missing per year between the two populations Use df 7 if needed 2 What would we conclude about H0 11 12 0 Explain Instructional Video example A company is interested in the value of a new instructional Video designed to reduce errors on the production line Six production line workers are randomly sampled and their monthly errors are recorded before and after watching the Video Because the same workers are measured twice the samples are not independent Errors before 12 9 0 5 4 3 Errors after 9 6 l 3 2 3 1 Calculate the mean cl and standard deviation Sd for the sample differences Instructional Video page 2 2 Find the 95 confidence interval for the average reduction in monthly errors after the workers watch the instructional Video Does the evidence suggest the company should have all production line workers watch the video Explain Animal bites example We have the following sample information about mail carriers in two cities and the proportion who have received animal bites City n bit 1 Cleveland 73 10 2 Philadelphia 80 16 Find the 95 confidence interval for the difference between proportions in the two cities Do we have evidence that the two population proportions differ Explain Leisure time example Random samples of females and males were asked if they have more leisure time now than they did ten years ago Gender n Answered yes 1 Female 50 44 2 Male 30 28 Are the population proportions the same for each gender Test at 1 005 F distributi0n example Assume the numerator df 2 and the denominator df 12 1 What is the upper tail probability if F 281 2 What is the upper tail probability if F 389 3 What is the upper tail probability if F 6 Filtration system example We have three assembly methods for a filtration system and we want to know which one is best We randomly assign ve workers to each method and count the number assembled in one week The summary data is provided below Method Sample size 5 5 5 Sample mean 62 66 52 The sample means are not equal Are the population means equal 1 Complete the AN OVA table based on the computer printout Analysis of Variance Table Response y Df Sum Sq Mean Sq F value PrgtF treatments 2 520 260000 91765 00038l8 Residuals l2 340 28333 AN OVA table Source ofvariation Sum of squares df Mean square F Treatments Error residuals Total Filtration system page 2 2 Conduct the appropriate hypothesis test at 1 005 Carefully show all ve steps Partial AN OVA table examples 1 Complete the AN OVA table no context Treatment A Sample size 4 5 5 Sample mean 310 413 360 440 AN OVA table Source of variation Sum of squares df Mean square Treatments 4369 Error residuals Total 9477 2 Complete the ANOVA table no context new example AN OVA table Source of variation Sum of squares df Mean square Treatments 35 Error residuals Total 94 Partial AN OVA page 2 3 Conduct the appropriate hypothesis test for the previous AN OVA table at 1 005 Carefully show all ve steps Sick days example An employer wants to know if there is a relationship between the age X of an employee and the number of sick days y taken each year Agex I182639485358 ISickdaysy16129562 Make the scatter diagram Persistence example An experimenter gave three subjects a very dif cult impossible task The age of the subject X and the number of attempts made to accomplish the task before giving up y were recorded This is a very small data set IAgeX 20 55 30 IAttemptsy 5 12 10 1 Make the scatter diagram 2 Find the sample mean and variance of X Persistence page 2 3 Find the sample mean and variance of y 4 Find the covariance of X and y Persistence page 3 5 Find the sample correlation coefficient Persistence example cont 1 2 3 Find b0 and b1 Use results from the previous lecture as needed Write the estimated SLR equation What is the estimated number of attempts for a 40yearold Sick days example cont Recall the data for this example Agex I18269485358 ISickdaysyl6129562 U We are also given the following summaiy information a 4033 y 833 2xl 902 123733 2yi y2 12933 206i 900 7 39167 1 Find 5 and 33 2 Find the sample covariance Sxy Sick days page 2 3 4 5 Find the sample correlation coefficient rxy Find b0 b1 and the estimated SLR equation How many sick days are expected for a 35yearold Homework Stat 204 Exam 3 Solutions to selected problems are located at the end of the document The problems assigned for each lab are due at the end of that lab meeting HW7 1 Consider the hypothesis test So 03 F H0 u 2 60 Ha u lt 60 Assume we have a sample of size n 85 and the population standard deviation is 11 Find the p value and state your conclusion for each of the following sample means Use 04 001 a i 59 b i 58 c i 57 d E 62 think At the start of 2006 Fortune magazine reported that Wall Street securities rms had recently paid the average employee 125500 in end of year bonuses moneycnncom We want to know if the mean year end bonus for an employee at Company X is different from the reported population mean of 125500 a Write down the appropriate null and alternative hypotheses b A sample of 35 employees at Company X has a mean year end bonus of 121000 Find the p value for this test if the population standard deviation is 19000 c State your conclusion at the 04 005 level of signi cance Consider the hypothesis test H0 2 ILL 47 Ha 1 7E 47 A sample of size 18 has a mean of 45 and a standard deviation of 35 a Find the test statistic c At 04 005 state your conclusion b Find a range for the p value using the 25 distribution table For the month of June in 2010 the US Department of Labor reported a national mean unemployment insurance bene t of 306 per week dolgov A researcher in Colorado suspected that sample data would provide evidence that the mean weekly bene t in her state was above the national level a Write down the appropriate null and alternative hypotheses to test her claim b A sample of 27 individuals from Colorado had a mean weekly unemployment insurance bene t of 349 and a standard deviation of 119 Find the p value for this test c State your conclusion at the 04 005 level of signi cance 5 a The Bureau of Labor Statistics reported that 123 of US workers belonged to unions in 2009 bls gov To test the claim that union membership was declining7 a researcher sampled 479 US workers in 2010 and found that 57 belonged to unions a Write down the appropriate null and alternative hypotheses to test his claim that union membership declined in 2010 b Find the p value for this test c State your conclusion at the 04 005 level of signi cance A researcher predicted that during the second week of June in 20117 less than 51 of US households watching television would watch the show Hawaii Five 0 A sample consistent with the results found by the Nielsen Company showed that of 25000 households watching television that week7 1225 watched Hawaii Five O nielsen com a For US households watching television that week7 nd the point estimate of the pro portion that watched Hawaii Five 0 b Test the claim at 04 005 Carefully show all ve steps END OF HW 7 pROBLEMS HW8 1 Consider the hypothesis test I tp14p20 Ha 3 1 i 2 739 0 The following information is from independent samples taken from the two populations As sume the population standard deviations are 51 and 827 respectively n i Sample 1 33 13 1 Sample 2 41 10 2 a Find the test statistic b Find the p value c At 04 0057 state your conclusion 2 ln 20097 an educational researcher believed the average teacher salary was higher in North Dakota than in South Dakota She gathered the following data from independent samples of teachers from each state l l State l n E s l 1 North Dakota 43 41030 7100 2 South Dakota 47 38540 6300 To study the effectiveness of a new commercial for their product7 a market research rm selected a sample of nine individuals Each individual was rst asked to rate their likelihood of purchasing the given product7 on a scale from 0 to 10 After watching the new commercial the same nine individuals were asked the same question At 04 0057 do the following data indicate that the mean rating increased after watching the commercial Carefully show all ve steps of the hypothesis test llndividualll 2 3 4 5 6 7 8 9 After 376879537 Before 7 5 8 5 5 4 u 4 0 In June 20117 a Gallup poll asked individuals from different regions of the country about their con dence in US banks gallupcom Of speci c interest for this problem is the proportion of respondents who had much con dence in US banks Data consistent with Gallup s results are given Sample Had quotmuchquot Region size confidence 1 East 980 2 West 1074 247 a Estimate the proportion from each region who had much77 con dence in US banks Which region had the larger proportion b Find the point estimate for the difference between the proportions of the two regional populations c Find the 95 con dence interval for the difference between the two population propor tions d At 04 0057 is it believable that the two population proportions are actually the same Explain ln 19987 The Washington Post reported demographic information about the Virginia lottery washingtonpost com Data consistent with the results about the proportion of males and females who play the lottery are given Sample Play the Gender size lottery Male 325 156 Female 250 85 a Find the point estimate of the proportion of males who indicate they play the lottery Find the point estimate for females b Find the point estimate for the difference between the two population proportions c At 95 con dence7 nd the margin of error for the estimate in d Find the 95 con dence interval for the difference between the two population propor tions END OF HW 8 pROBLEMS HW9 1 Complete the following ANOVA table For this design we have samples of size seven from each of ve different populations Response y Df Sum Sq Mean Sq F value 3 Treatments 59 Residuals Total 2059 What is the p value E0 For the previous problem7 a Write down the implied null and alternative hypotheses b At 04 0057 state your conclusion In March 20097 Motor Trend conducted a road test comparison of the 2010 Chevrolet Camaro SS7 the 2009 Dodge Challenger RT7 and the 2010 Ford Mustang GT motortrend com One of the criteria measured was the time required to travel one quarter mile from a stand still For purposes of this problem7 assume six cars of each brand were tested by the same driver and the standing quarter mile time in seconds was recorded for each car Results consistent with Motor Tremfs analysis are provided OJ Chevy 129 131 131 130 130 129 Dodge 138 134 136 135 136 137 Ford 133 135 135 136 137 134 This is the ANOVA table for the data Response Acceleration Df Sum Sq Mean Sq F value PrgtF Treatments 2 1240 0620 3875 1187e06 Residuals 15 0240 0016 a Find the Total df b Find the Total SS c Find the point estimate for the mean standing quarter mile time for each of the three cars d At 04 0057 carefully show all ve steps of the hypothesis test for signi cant difference in the population means of standing quarter mile times e Is it likely that the three cars have the same mean standing quarter mile times Explain NOTE Motor Trend selected the Chevy Camaro as the Prime Pony of the 21st Century77 This has no bearing on the calculations for the problem 4 U The following ve observations were taken for two variables 2 3 5 7 11 M 4 6 10 8 12 a Make a scatter diagram for the data b Does the scatter diagram indicate a possible relationship between z and 3 Explain c Find the sample covariance Use the alternate formula d Find the sample correlation coef cient Interpret Nielsen Media Research provides two values for each television show The rating measures the percentage of viewers with televisions watching the show7 and the share measures the percentage of viewers watching the show among those viewers whose televisions are on during that time slot The following data show the rating and share for a sample of six of the 45 top rated programs of all time wikipedia com 43 44 46 48 51 60 62 63 71 67 71 77 Rating m Share y END OF HW 9 pROBLEMS Additional problems These problems will niot be collected for a grade but the material they cover m be on the exam 1 Consider the ve observations for two variables x and y E0 gal89567 34129251418 A glider car is pushed along a nearly frictionless track and is released at the 10 cm mark on the track At the time of release a stopwatch is started At two second intervals the location of the car is recorded The data appear in the following table Time m seconds 2 4 6 8 10 Location y cm 128 169 222 253 292 A a Make a scatter diagram for Location versus Time U Does the scatter diagram indicate a possible relationship between Time and Location Explain Find the estimated regression equation by computing 0 and 1 A 0 VV In this problem o represents the location of the car when the stopwatch was started How does 0 compare to the true value A D V The slope l represents the constant velocity of the car in cmsec What is the esti mated velocity of the car based on our data A l h V Use the estimated regression equation to nd the predicted Location after ve seconds Selected solutions HW 7 l a 02005 fail to reject H0 b 00465 fail to reject H0 c 00060 reject H0 d 09535 fail to reject H0 3 a t 7242 b 002 lt pvalue lt 005 15 1 2 a 005 3 z 7144 4 pvalue 00749 5 fail to reject H0 Selected solutions HW 8 l a z 186 b pvalue 00628 c fail to reject H0 2 a H001702 0andHaluliluggt0 t 175 c 0025 lt pvalue lt 005 d reject H0 4 151 020 and 152 023 151 7152 c 70066 lt 191 7192 lt 0006 d yes 0 is in the con dence interval Selected solutions HW 9 1 Response y U H Sum Sq Mean Sq F value Treatments 4 593 1483 303 Residuals 30 1466 489 Total 34 205 9 0025 lt pvalue lt 005 a 17 b 148 C EChevy 130 EDodge 136 ipord 135 d 1 H0 ucmvy uDOdge upgrd and Ha at least one mean differs 2 Oz 005 3 F 3875 4 p value 11875 7 06 1187 X 10 6 0000001187 5 reject H0 e no the evidence supports the hypothesis that at least one mean differs b yes there appears to be a positive possibly linear relationship c 10 d 0884 indicates a positive linear relationship Solutions for additional problems b yes there appears to be a negative possibly linear relationship c be 394121 734 g 394 7 34x 13 394 7 346 19 We b yes there appears to be a strong7 positive linear relationship c be 892 1 206 37 892 206m or Lo ition 892 206Time d estimate 892 cm true 10 am e estimated velocity 206 cmsec f g 892 2065 1922 cm
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