Answer to Question #345998 in Statistics and Probability for Banjo

Question #345998

A random sample of 25 brand A cigarettes showed an average nicotine content of 5 milligrams, while a sample of 40 brand D cigarettes showed average nicotine of 4.8 milligrams. If the standard deviation of nicotine is 1.6 milligrams, would you say that brand D has a lesser nicotine content? Use a 0.01 level of significance. Assume the distribution of nicotine content to be normal

1
Expert's answer
2022-05-30T15:57:14-0400

The following null and alternative hypotheses need to be tested:

"H_0:\\mu_1\\le\\mu_2"

"H_0:\\mu_1>\\mu_2"

This corresponds to a right-tailed test, and a z-test for two means, with known population standard deviations will be used.

Based on the information provided, the significance level is "\\alpha = 0.01," and the critical value for a right-tailed test is "z_c = 2.3263"

The rejection region for this right-tailed test is "R = \\{z: z > 2.3263\\}."

The z-statistic is computed as follows:


"z=\\dfrac{\\bar{X}_1-\\bar{X}_2}{\\sqrt{\\sigma_1^2\/n_1+\\sigma_2^2\/n_2}}"

"=\\dfrac{5-4.8}{\\sqrt{1.6^2\/25+1.6^2\/40}}\\approx0.4903"

Since it is observed that "z = 0.49 \\le z_c = 2.326," it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is "p=P(Z>0.4903)=0.311961," and since "p=0.311961>0.01," it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean "\\mu_1"is greater than "\\mu_2," at the "\\alpha = 0.01" significance level.

Therefore, there is not enough evidence to claim that brand D has a lesser nicotine content, at the "\\alpha = 0.01" significance level.


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