statistics math question

Question

A food safety guideline is that the mercury in fish should be below 1 part per million​ (ppm). Listed below are the amounts of mercury​ (ppm) found in tuna sushi sampled at different stores in a major city. Construct a

90​%

confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna​ sushi?

0.55  

0.71  

0.10  

0.98  

1.35  

0.53  

0.91

 

Answer 1

1.) First, figure out what the question is asking you to do.

1. Find a 90 percent confidence interval (CI) for the population mean. (note: that your instructor is asking you specifically about the population mean and NOT about the difference between the population mean and the threshold value of mercury. Working with this hypothesized difference is another approach to solving the problem.)
2. Determine if the population mean is greater than the allowable value of mercury.

2.) Identify the information given in the problem:

1. sample of 7 observations
2. sample size = 7

3.) Decide which test statistic you will use. For this type of problem will not need to calculate the test statistic, itself, but only need to find its "critical value."

1. Do you know the population standard deviation? No
2. Is your sample size large enough to use the sample standard deviation as a substitute for the population standard deviation? No, N = 7. You need at least 25 - 30 observations to "pretend" you know the population standard deviation.
3. So Z tests are out. You will need a t test.
4. You are building a two-sided confidence interval. (The question, technically, asks a "one-tailed" question, i.e., are the mercury levels HIGHER than a given value, but you will be working with two-sided confidence intervals for most of your courses).
5. Find the critical value for t
6. You want a two-tailed significance of 1 - .90 = ..10 for the two-sided 99 percent confidence interval
7. You have N - 1 = 7 - 1 = 6 degrees of freedom for this t test. (one sample t)
8. There should be a t-test table in the back of your book, or you can use Excel's T.INV.2T functions. T.INV.2T(.10, 6) will give you your critical t value. You can read more about this excel function at , https://www.excelfunctions.net/excel-t-inv-2t-function.html#:~:text=Function%20Description-,The%20Excel%20T.,on%20small%20sample%20data%20set
9. calculate the mean of your sample
10. Calculate your standard error of the mean, = (sample SD)/sqrt(N)
11. Construct your confidence interval: CI_90 = sample mean +- t_critical*SE
12. Decide if the 1 part per million value for mercury contamination is inside or outside the confidence interval.
13. If it is inside the interval, then you do NOT have the evidence to conclude that there is too much mercury in the sushi.
14. If it is outside the interval AND the sample mean is greater than 1 part per million, you may conclude that the mean amount of mercury is HIGHER than 1 part per million
15. (In the unlikely event that it is outside the interval AND the sample mean is less than 1 part per million, you may conclude that the amount of mercury is LOWER than 1 part per million).

Hope this helps you solve this problem and many others like it.