A case-control (or retrospective) study was conducted to investigate a relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in a crash. Results are given in the accompanying table. Using a 0.01significance level, test the claim that injuries are independent of helmet color.Color of HelmetBlackWhiteYellowRedBlueControls (not injured)Cases (injured orkilled)488367281619521311476846Click here to view the chi-square distribution table.Identify the null and alternative hypotheses. Choose the correct answer below.OA. Ho: Injuries and helmet color are independentH: Injuries and helmet color are dependentO B. Ho: Whether a crash occurs and helmet color are dependentH1: Whether a crash occurs and helmet color are independentO C. Ho: Injuries and helmet color are dependentH: Injuries and helmet color are independentO D. Ho: Whether a crash occurs and helmet color are independentH,: Whether a crash occurs and helmet color are dependentCompute the test statistic.(Round to three decimal places as needed.)

One type of parametric test which used to measure whether there is any association between the two categorical variable with various level is known as chi square test. This test also used to measure independence of attribute and goodness of fit. It is denoted as χ2. It is calculated by solving the two values, they are observed and expected value and their formula is defined as : χ2=∑O-E2EGiven : a) The relationship between the colors of helmet and the motor drivers injured or killed is given as : Black White Yellow Red Blue Total Not injured 488 367 28 161 95 1139 Injured 213 114 7 68 46 448 Total 701 481 35 229 141 1587 Claim : Injuries are independent of helmet color Null hypothesis : Injuries are independent of helmet color Alternate hypothesis : Injuries are dependent of helmet color Hence option A is correct. Incorrect options : Option B - Have to compare injuries and the helmet color not crash occurrence Option C - Claim is not in alternate hypothesis while testing association Option D - Have to compare injuries and the helmet color not crash occurrence and also Claim is not in alternate hypothesis while testing associationb) Expected value is calculated using the formula : Expected value=Row total×Column totalGrand total Now using the observed values, the expected values will be : Black White Yellow Red Blue Total Not injured 1139×7011587=503.112 1139×4811587=345.217 1139×351587=25.120 1139×2291587=164.355 1139×1411587=101.197 1139 Injured 448×7011587=197.888 448×4811587=135.783 448×351587=9.880 448×2291587=64.645 448×1411587=39.803 448 Total 701 481 35 229 141 1587 Now, using the observed and expected values, the test statistic will be : Observed Expected O-E O-E2 O-E2E 488 503.112 -15.112 228.3725 0.45392 213 197.888 15.112 228.3725 1.154049 367 345.217 21.783 474.4991 1.374495 114 135.783 -21.783 474.4991 3.49454 28 25.12 2.88 8.2944 0.330191 7 9.88 -2.88 8.2944 0.839514 161 164.355 -3.355 11.25602 0.068486 68 64.645 3.355 11.25603 0.174121 95 101.197 -6.197 38.40281 0.379486 46 39.803 6.197 38.40281 0.964822 Total 9.233624 In the table, the expected value and the observed value are used to estimate the chi square. From this table, the chi square value is 9.234The degree of freedom : df=r-1c-1=2-15-1=1×4=4 Level of significance is 0.01 The critical value at 0.01 level of significance with 4 degree of freedom is 13.277 Since the test statistic value 9.234 is less than critical value, the null hypothesis fail to be rejected. Hence it is concluded that Injuries are independent of helmet color.