Question

Carleton Chemical claims that they can produce an average of more than 800 tons of meladone

per week. A random sample of 36 weeks of production yielded a sample mean of 823 tons, with
a standard deviation of 79.8 tons.

Does the sample data provide sufficient evidence to support the claim
made by Carleton Chemical? Use a significance level of α = .05.

Answer 1

t=1.729

`p_v =P(t_((35))>1.729)=0.0463`

Explanation:

1) Data given and notation

`\bar X=823` represent the mean production for the sample

`s=79.8` represent the sample standard deviation for the sample

`n=36` sample size

`\mu_o =800` 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)

Part a: State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean production is higher than 800 tons, the system of hypothesis would be:

Null hypothesis:
`\mu \leq 800`

Alternative hypothesis:
`\mu > 800`

If we analyze the size for the sample is > 30 but 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".

Part b: Calculate the statistic

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

`t=(823-800)/((79.8)/(√(36)))=1.729`

Part c: P-value

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

`df=n-1=36-1=35`

Since is a one side test the p value would be:

`p_v =P(t_((35))>1.729)=0.0463`

Part d: 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 reject the null hypothesis, so we can conclude that the production its significant higher compared to the value of 800 tons at 5% of signficance.