Answer to Question #350644 in Statistics and Probability for Matt

Question #350644

A certain group of welfare recipients receives relief goods with a mean amount of Php 500.00 per week. A random sample of 75 recipients is survived and found that the mean amount of relief goods they received in a week is Php 600 and a standard decision of Php. 50.00. Test the claim at 1% level of significance is not Php 500.00 per week and assume that the population is approximately normally distributed. Which of the following is correct null and alternative hypothesis, in symbol?


1
Expert's answer
2022-06-15T13:04:38-0400

The following null and alternative hypotheses need to be tested:

"H_0:\\mu=500"

"H_1:\\mu\\not=500"

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=74" and the critical value for a two-tailed test is "t_c =2.643913."

The rejection region for this two-tailed test is "R = \\{t:|t|>2.643913\\}."

The t-statistic is computed as follows:



"t=\\dfrac{\\bar{x}-\\mu}{s\/\\sqrt{n}}=\\dfrac{600-500}{50\/\\sqrt{75}}=17.3205"


Since it is observed that "|t|=17.3205>2.643913=t_c," it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, "df=74" degrees of freedom, "t=17.3205" 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 500.00, at the "\\alpha = 0.01" significance level.


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