Question

From Statistics and Data Analysis from Elementary to Intermediate by Tamhane and Dunlop, pg 265. A thermostat used in an electrical device is to be checked for accuracy of its design setting of 200◦F. Ten thermostats were tested to determine their actual settings, resulting in the following data: 202.2 203.4 200.5 202.5 206.3 198.0 203.7 200.8 201.3 199.0 Perform the t-test to determine if the mean setting is different from 200◦F. Use α = 0.05

Answer 1

Answer:

$t=\frac{201.77-200}{\frac{2.41}{\sqrt{10}}}=2.32$    

The degrees of freedom are given by:

$df=n-1=10-1=9$  

The p value for this case is given by:

$p_v =2*P(t_{(9)}>2.32)=0.0455$    

For this case since the p value is lower than the significance level we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significantly different from 200 F.

Step-by-step explanation:

Information given

data: 202.2 203.4 200.5 202.5 206.3 198.0 203.7 200.8 201.3 199.0

We can calculate the sample mean and deviation with the following formulas:

$\bar X= \frac{\sum_{i=1}^n X_i}{n}$

$\sigma=\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}$

$\bar X=201.77$ represent the sample mean    

$s=2.41$ represent the sample standard deviation    

$n=10$ sample size    

$\mu_o =200$ represent the value that we want to test    

$\alpha=0.05$ represent the significance level for the hypothesis test.    

t would represent the statistic

$p_v$ represent the p value for the test

Hypothesis to test

We want to determine if the true mean is equal to 200, the system of hypothesis are :    

Null hypothesis:$\mu = 200$    

Alternative hypothesis:$\mu = 200$    

The statistic for this case is given by:

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

The statistic is given by:

$t=\frac{201.77-200}{\frac{2.41}{\sqrt{10}}}=2.32$    

The degrees of freedom are given by:

$df=n-1=10-1=9$  

The p value for this case is given by:

$p_v =2*P(t_{(9)}>2.32)=0.0455$    

For this case since the p value is lower than the significance level we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significantly different from 200 F.