Question

Our environment is very sensitive to the amount of ozone in the upper atmosphere. The level of ozone normally found is 4.6 parts/million (ppm). A researcher believes that the current ozone level is not at the normal level. The mean of 860 samples is 4.5 ppm. Assume the standard deviation is known to be 1.1. Does the data support the claim at the 0.02 level? Step 5 of 5: Enter the conclusion.

Answer 1

`z=(4.5-4.6)/((1.1)/(√(860)))=-2.666`

`p_v =2*P(Z<-2.666)=0.0077`

If we compare the p value and the significance level given
`\alpha=0.02` 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 true mean is different from 4.6 at 2% of signficance.

Explanation:

Data given and notation

`\bar X=4.5` represent the sample mean

`\sigma=1.1` represent the population standard deviation

`n=860` sample size

`\mu_o =4.6` represent the value that we want to test

`\alpha=0.02` represent the significance level for the hypothesis test.

z 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 mean is equal or not to 4.6, the system of hypothesis would be:

Null hypothesis:
`\mu = 4.6`

Alternative hypothesis:
`\mu \\eq 4.6`

If we analyze the size for the sample is > 30 and we know the population deviation so is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:

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

z-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:

`z=(4.5-4.6)/((1.1)/(√(860)))=-2.666`

P-value

Since is a two sided test the p value would be:

`p_v =2*P(Z<-2.666)=0.0077`

Conclusion

If we compare the p value and the significance level given
`\alpha=0.02` 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 true mean is different from 4.6 at 2% of signficance.