Question

A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 445 gram setting. It is believed that the machine is underfilling the bags. A 13 bag sample had a mean of 443 grams with a standard deviation of 30. Assume the population is normally distributed. A level of significance of 0.05 will be used. Specify the type of hypothesis test.

Answer 1

`t=(443-445)/((30)/(√(13)))=-0.240`

`p_v =P(t_((12))<-0.240)=0.407`

If we compare the p value and the significance level given
`\alpha=0.05` we see that
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL reject the null hypothesis, so we can't concluce that the true mean is significantly less than 445 grams at 5% of significance.

Explanation:

Data given and notation

`\bar X=443` represent the sample mean

`s=30` represent the sample standard deviation

`n=13` sample size

`\mu_o =445` represent the value that we want to test

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

t would represent the statistic (variable of interest)

`p_v` represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the machine is underfilling the bags , the system of hypothesis would be:

Null hypothesis:
`\mu \geq 445`

Alternative hypothesis:
`\mu < 445`

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic

We can replace in formula (1) the info given like this:

`t=(443-445)/((30)/(√(13)))=-0.240`

P-value

The first step is calculate the degrees of freedom, on this case:

`df=n-1=13-1=12`

Since is a one lower tailed test the p value would be:

`p_v =P(t_((12))<-0.240)=0.407`

Conclusion

If we compare the p value and the significance level given
`\alpha=0.05` we see that
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL reject the null hypothesis, so we can't concluce that the true mean is significantly less than 445 grams at 5% of significance.