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In the past, the mean age of consumers of a product was 37 years. Recently the product has been drastically changed. In order to determine whether there has been a change in the mean age of the product consumers, a sample of 16 consumers is selected. The mean age in the sample is 31 years and the standard deviation in the sample is 10 years. Let μ = the mean age of the current product consumers.

At a 1% significance level, the null hypothesis is rejected if:_________.

User Mixalis
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Answer:


t=(31-37)/((10)/(√(16)))=-2.4

The degree of freedom are:


df=n-1=16-1=15

We can calculate the critical value for this case with the degrees of freedom ,we can find a critical value in the t distribution with 15 degrees of freedom who accumulate 0.005 of the area on each tail of the distribution and we got:


t_(crit)=\pm 2.947

And we will reject the null hypothesis is the statistic for this case satisfy this condition:


|t_(calculated)| > |t_(critical)|= 2.947

Explanation:

Information given


\bar X=31 represent the sample mean


s=10 represent the sample standard deviation


n=16 sample size


\mu_o =37 represent the value to verify


\alpha=0.01 represent the significance level

t would represent the statistic


p_v represent the p value for the test

System of hypotheis

We want to verify if the actual mean age of consumers of a product was 37 years and the system of hypothesis are:

Null hypothesis:
\mu = 37

Alternative hypothesis:
\mu \\eq 37

We can use the following statistic for this case:


t=(\bar X-\mu_o)/((s)/(√(n))) (1)

Replacing the info given we got:


t=(31-37)/((10)/(√(16)))=-2.4

The degree of freedom are:


df=n-1=16-1=15

We can calculate the critical value for this case with the degrees of freedom we can find a critical value in the t distribution with 15 degrees of freedom who accumulate 0.005 of the area on each tail of the distribution and we got:


t_(crit)=\pm 2.947

And we will reject the null hypothesis is the statistic for this case satisfy this condition:


|t_(calculated)| > |t_(critical)|= 2.947

User Ritu Suman Mohanty
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8.1k points

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