Question

A special cable has a mean breaking strength of 1131 pounds. The standard deviation of the population is 333 pounds. A researcher selects a sample of 30 cables with a mean of 931 pounds. Can one reject the claim that the mean breaking strength is less than 1131 pounds? (Use the 0.05 level of significance)​

Answer 1

`z=(931-1131)/((333)/(√(30)))=-3.290`

Now we can calculate the p value using the alternative hypothesis:

`p_v =P(z<-3.290)=0.0005`

Since the p value is lower than the significance level of 0.05 we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significantly less than 1131

Explanation:

Information given

`\bar X=931` represent the sample mean

`\sigma=333` represent the population standard deviation

`n=30` sample size

`\mu_o =1131` represent the value to check

`\alpha=0.05` represent the significance level

z would represent the statistic

`p_v` represent the p value

System of hypothesis

We want to verify if the true mean is less than 1131, the system of hypothesis would be:

Null hypothesis:
`\mu \geq 1131`

Alternative hypothesis:
`\mu <1131`

The statistic is given by:

`z=(\bar X-\mu_o)/((\sigma)/(√(n)))` (1)

Reaplacing we got:

`z=(931-1131)/((333)/(√(30)))=-3.290`

Now we can calculate the p value using the alternative hypothesis:

`p_v =P(z<-3.290)=0.0005`

Since the p value is lower than the significance level of 0.05 we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significantly less than 1131