Math 52 Answers to TEST 4 - Santa Monica College …

1 math 52 Answers to TEST 4 Note: It is assumed that all the populations in the problems are normally distributed.(1) Find the Critical value(s) in (a) (d) andp-value in (e) for the following hypothesis testing:(a) a left-tailed test about the population mean with given, the significance level is = (when is used, the test statistic is azscore:) = (or )(b) a two-tailed test about the population mean , unknown, withn= 22 and = (when is not given andsis used, the test statistic is atscore:)= t 2, df= ,21= (c) a right-tailed test about population variance 2, withn= 20 and = (the test statistic is a 2score:) = 2 , df= ,19= (d) When the claim is 5 and is not given, using a sample withn= 30 and = claim is: 5 = theH1is : <5 = a left-tailed test.

2 (when is not given andsis used, the test statistic is atscore,)= t , df= ,29= (e) What is thep-value in a left tailed test if the isz= a left-tailed test, thep-value is the left hand side area under the density curve,from Table A-2 = p= (2) (A hypothesis test for a given claim will be conducted.)The claim is: The mean fuel consumption rate for a certain car is no more than 19 (mi/gal).(a) Express the claim in mathematical form: = 19(b) What is the null hypothesisH0and the alternative hypothesisH1;H0is: = 19 andH1is: >19(c) (Identify the test to be two-tailed, left-tailed or right-tailed:) = a right-tailed test(d) If you fail to reject the null hypothesisH0according to your sample, what final conclusionwill you make on the claim?

3 (sinceH0is with the claim)= There is No sufficient sample evidence to warrant the rejection of the hypothesis testing to test the given claim: (show work step by step.)(3) Test the claim that a population mean is more than 420. You have a random sample of 15 itemsfor which the sample mean x= and the sample standard deviations= 44. Use a significance level = (using the traditional method)the claim is: >420H0: = 420,H1: >420, Type of test:right tailed, = Test Statistic is:t= x s n= 42044 15 Critical Value is:t , df= ,14= Initial conclusion is: (the test statistic in NOT in the critical region) = failed to rejectH0,the Final Conclusion is:There is NOsufficient evidence to supportthe claim that the population mean is more than 420.

4 (4) Before purchasing a large amount of light bulbs from a certain company, a random sample of 8 lightbulbs produced by that company is selected. The lifetime of each bulb is tested (measured in hours). It wasfound that x= ands= acceptable standard deviation in life times of the light bulbs is 20 (hours) (the claim), otherwise,the light bulbs from that company should not be purchased. Use = to test the claim 20. (usetraditional method.)the claim is: 20H0: = 20,H1: >20, Type of test:right tailed, = Test Statistic is: 2=(n 1)s2 2=(8 1) Critical Value is: 2 , df= ,7= Initial conclusion is: (the test statistic in NOT in the critical region) = failed to rejectH0,the Final Conclusion is:There is NOsufficient evidence to rejectthe claim that the 20 (hours).

5 (5) A supplier of disks claims that no more than 1% of the disks are defective. In a random sampleof 600 disks, it is found that 3% are defective. At the significance level, test the supplier s claim thatno more than 1% are defective. (use both traditional method andP-value method.)the claim is:p :p= ,H1:p > , Type of test:right tailed, = Test Statistic is:z= p p p qn= ( )( )600 Critical Value is:z = is:p= (the right-hand side area ofz= and under the density curve.)the Initial conclusion is:by Traditional method: (the test statistic is in the critical region) = rejectH0with = method: sincep < = rejectH0with = Final Conclusion is: There issufficient evidence to rejectthe claim that no more than 1% of thedisks are defective.

6 (that is, the defective rate is significantly more than 1.)(6) A cereal company claims that the mean weight of the cereal in its packets is 14oz. The weights (inounces) of the cereal in a random sample of 8 of its cereal packets are listed , , , , , , , ,Test the claim at the significance level, use traditional method.(to save time, the following sample statistics are given:n= 8, x= , and x2= )by calculation, = x= ,s= ,the claim is: = 14H0: = 14,H1: 6= 14, Type of test:two tailed, = Test Statistic is:t= x s n= 8 Critical Value is: t 2, df= ,7= Initial conclusion is: (the test statistic is NOT in the critical region) = failed to rejectH0,the Final Conclusion is:There is NOsufficient evidence to rejectthe claim that the mean weight of the package is 14 oz.

7 (7) Find (a) the test statistics ( ) and (b)P-value used to test the claim:p1=p2(comparing two population proportions).Sample 1:n1= 140 andx1= 15;Sample 2:n2= 217 andx2= 21(a) p1=15140= , p2=21217= , p=x1+x2n1+n2=15+21140+217= ,the test statistic is:z=( p1 p2) (p1 p2) ( p)( q)(1n1+1n2)=( ) (0) ( )( )(1140+1217) (b) this is a two-tailed test, thep-value is twice of the outside area of the test statistic= p= 2( ) = (8) Two independent random samples are drawn from two populations as follows:n1= 75, x1= ;n2= 51, x2= 4s2= (a) Construct a 99% confidence interval for the difference between the means of the twopopulations: ( 1 2), and (b) interpret the result.(a) (find the margin of error first:) = E=t 2, df s21n1+s22n2= + 99% confidence interval (on these two samples) is:( x1 x2) E= ( 4) = <( 1 2)< (b) (interpretation of the result:)With 99% confidence, the difference ( 1 2) is a positive number since it is within ( , ),= ( 1 2)>0 = 1> 2.

8 That is: the 1st population mean is significantly greaterthan the 2nd population mean.(9) Two machines fill 32-oz cartons with cereal. The quality control manager claims that cartons filledby machine 1 have more cereal than those filled by machine 2. Samples from each line were taken and thefollowing results were 1 (from machine 1)n1= 38 x1= (oz),s1= (oz);Sample 2 (from machine 2)n2= 45 x2= (oz),s2= (oz);Use the two samples to test the claim that population mean for machine 1 is larger than that ofmachine 2 with = Claim is 1> 2or ( 1 2)>0(1)H0is 1= 2,H1is 1> 2, Type of test isright tailed;(2) = (3) Test statistic:t=( x1 x2) ( 1 2) s21n1+s22n2=( ) (0) + (4) The Critical value is:t , df= ,37 (5) Initial Conclusion: the test statistic is in the critical region = rejectH0with = (6) Final Conclusion: There issufficient evidence to supportthe claim that the population meanfrom machine 1 is greater than that from machine 2.

9 (10) The following are costs of advertising (in thousnads of dollars) and the numbers of products sold(in thousands):Costx9 2 3 4 2 5 9 10products soldy85 52 55 68 67 86 83 81 For this data set, the following statistics are given: x= 44, y= 577, x2= 320, y2= 42913, xy= 3427,(a) Find the linear correlation coefficientr,r=n xy ( x)( y) n( x2) ( x)2 n( y2) ( y)2=8(3427) (44)(577) 8(320) (44)2 8(42913) (577)2 (b) Is there a significant linear correlation existing betweenxandy? (use = )In table A-6, withn= 8, the critical value is ( = ),|r|= > = There exists significant linear correlation in betweenxandy( = ).(c) Find the linear regression equation that best fits the paired data, (keep 3 decimal places inb0andb1)(First, calculate the slope b1) = b1=n( xy) ( x)( y)n( x2) ( x)2=8(3427) (44)(577)8(320) (44)2= (Then, find they-interceptb0) = b0= y b1 x= (5778 ( )(448)) = the linear regression equation is: y=b0+b1x= y= + (d) Make a scatterplot of the 8 pairs and graph the linear regression line,-6qqqqqqqqqqqq246810x5060708090yvvv vvvvv