Answer to Question #210265 in Statistics and Probability for Michell

Question #210265

A seller claimed that her lip tint has a mean organic content of 90%. Arrival seller ask 60 users of that lip tint and found that it has a mean organic content of 85% with a standard deviation of 5%. Test the claim at 1% level of significance and assume that the population is approximately normally distributed


1
Expert's answer
2021-06-24T18:34:27-0400

The following null and alternative hypotheses need to be tested:

"H_0:\\mu=90"

"H_1:\\mu\\not=90"

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is "\\alpha=0.01," 

"df=n-1=60-1=59" ​​degrees of freedom, and the critical value for a two-tailed test is"t_c=2.661759."

The rejection region for this two-tailed test is "R=\\{t:|t|>2.661759\\}."

The t-statistic is computed as follows:


"t=\\dfrac{\\bar{x}-\\mu}{s\/\\sqrt{n}}=\\dfrac{85-90}{5\/\\sqrt{60}}\\approx-7.7460"

Since it is observed that "|t|=7.7460>2.661759=t_c," it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean "\\mu" is different than 90, at the "\\alpha=0.01" significance level.


Using the P-value approach: The p-value for two-tailed, "\\alpha=0.01, df=59,"

"t=-0.10396" is "p=0," and since "p=0<0.01=\\alpha,"it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean "\\mu" is different than 90, at the "\\alpha=0.01" significance level.


Therefore, there is enough evidence to claim that the her lip tint has a mean organic content different than 90%, at the "\\alpha=0.01" significance level.



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