Question

The following prices, in dollars, of 7.5-cubic-foot refrigerators were recorded from a random sample. 314 305 344 283 285 310​ 383​ 285​ 300​ 300 A consumer organization reports that the mean price of 7.5-cubic-foot refrigerators is greater than $300. Do the data provide convincing evidence of this claim? Use the α = 0.05 level of significance and assume the population is normally distributed.

Answer 1

Answer:

$t=\frac{310.9-300}{\frac{31.09}{\sqrt{10}}}=1.109$      

The degrees of freedom are given by:

$df=n-1=10-1=9$  

And the p value would be given by:

$p_v =P(t_{9}>1.109)=0.148$  

Since the p value is higher than the significance level of 0.05 we don't have enough evidence to conclude that the true mean is significantly higher than $300 because we FAIL to reject the null hypothesis.

Step-by-step explanation:

Information given

314 305 344 283 285 310​ 383​ 285​ 300​ 300

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

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

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

$\bar X=310.9$ represent the sample mean      

$s=31.09$ represent the standard deviation for the sample      

$n=10$ sample size      

$\mu_o =300$ represent the value to test

$\alpha=0.05$ represent the significance level

t would represent the statistic

$p_v$ represent the p value

Hypothesis to test

We want to verify if the true mean is greater than 300, the system of hypothesis would be:      

Null hypothesis:$\mu \leq 300$      

Alternative hypothesis:$\mu > 300$      

The statistic is given by:

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

Replacing the info given we got:

$t=\frac{310.9-300}{\frac{31.09}{\sqrt{10}}}=1.109$      

The degrees of freedom are given by:

$df=n-1=10-1=9$  

And the p value would be given by:

$p_v =P(t_{9}>1.109)=0.148$  

Since the p value is higher than the significance level of 0.05 we don't have enough evidence to conclude that the true mean is significantly higher than $300 because we FAIL to reject the null hypothesis.