Question

A lumber company is making boards that are 2564.0 millimeters tall. If the boards are too long they must be trimmed, and if the boards are too short they cannot be used. A sample of 21 is made, and it is found that they have a mean of 2567.0 millimeters with a variance of 121.00. A level of significance of 0.1 will be used to determine if the boards are either too long or too short. Assume the population distribution is approximately normal. Find the value of the test statistic. Round your answer to three decimal places.

Answer 1

Answer:

$t=\frac{2567-2564}{\frac{11}{\sqrt{21}}}=1.250$    

The degrees of freedom are given by:

$df=n-1=21-1=20$  

the p value for this case would be given by:

$p_v =2*P(t_{(20)}>1.250)=0.2113$  

For this case we see that the p value is higher than the significance level so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is not significantly different from 2564 mm

Step-by-step explanation:

Information given

$\bar X=2567$ represent the mean height for the sample  

$s=\sqrt{121}= 11$ represent the sample standard deviation

$n=21$ sample size  

$\mu_o =2564$ represent the value that we want to test

$\alpha=0.1$ represent the significance level for the hypothesis test.  

t would represent the statistic

$p_v$ represent the p value

Hypothesis to test

We want to check if the true mean is equal to 2564 mm, the system of hypothesis would be:  

Null hypothesis:$\mu = 2564$  

Alternative hypothesis:$\mu \neq 2564$  

The statistic is given by:

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$  (1)  

Replacing we got:

$t=\frac{2567-2564}{\frac{11}{\sqrt{21}}}=1.250$    

The degrees of freedom are given by:

$df=n-1=21-1=20$  

the p value for this case would be given by:

$p_v =2*P(t_{(20)}>1.250)=0.2113$  

For this case we see that the p value is higher than the significance level so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is not significantly different from 2564 mm