The Colgate company believes its toothpaste is more effective than its leading r

Question

The Colgate company believes its toothpaste is more effective than its leading rival, Aim, in fighting tooth decay. A small scale clinical trial was held with 26 patients using Colgate and 22 using Aim. The results showed that the average and variance of the number of cavities for Colgate was 9.9 and 12.35, respectively, with the results for Aim being 11.5 and 10.42.

It is assumed that the population variances are the same and the distribution of the number of cavities for both groups closely approximates a normal distribution. Use alpha = 0.10.

1. If the hypotheses are to be written in the form of the average of Aim minus the average of Colgate, what is the direction of the alternative hypothesis used to test Colgate's belief. Type the letters gt (greater than), ge (greater than or equal to), lt (less than), le(less than or equal to) or ne (not equal to) as appropriate in the box.____________
2. Determine the standard error for the difference between means, reporting your answer to three decimal places.________
3. Calculate the test statistic, reporting your answer to two decimal places.__________
4. Is the null hypothesis rejected for this test? Type yes or no.____________
5. Regardless of your answer for 4, if the null hypothesis was rejected, would Colgate's belief appear to be correct at the 10% level? Type yes or no._________

Explanation / Answer

1) Claim is Colgate toothpaste is more effective than its leading rival, Aim

Null Hyp: H0: muc = mua

Alt Hyp: Ha: mua - muc < 0

Ans) lt

2) Std Error SE = sqrt[(s12/n1) + (s22/n2)]

                          = sqrt[(12.35/26) + (10.42/22)] = 0.974

3) Test stat = [ (x1 - x2) ] / SE = (11.5 - 9.9)/0.974 = 1.643

4) Degrees of Freedom DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }

DF = 46

alpha = 0.10

Critical value for test stat at 0.10 sig level and df = 46 is -1.3

Rejection region is t < -1.30

Since test stat does not lie in the rejection region we fail to reject the Null

Ans) No

5) No

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