Question

John works in the local waste station and notes that on any given day 1500 tons of waste, with a standard deviation of 500 tons is being received in these types of facilities across the city. John however, has become more quantitatively aware and finds that on average at his tip that he receives 1700 tons of waste per day. John would like to know if the amount of waste he receives is significantly more than the other tips.

a) To test Johns question what test would you use and why? Justify your answer.

b) What level of significance would be applicable for this test and why?

c) Conduct the hypothesis test using the critical value approach.

d) Construct a table using a significance level of 0.5, 1.0, 2.5, 5 and 10% and investigate how the conclusion would change as a result of changing the significance value.

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Answer 1

a) To test johns question, Critical Value of Z test i.e One tailed test could be used because:-

We want to test in only one direction here whether its significant , so the one tailed test is preferred here. Also with standard deviation and Mean given, could go for Z test easily.

b) Since level of significance isn't explicitly given in this problem, its default that we can use the significance to be 5% or 0.05

c)Lets solve it in steps:-

Step 1:- State the hypothesis, Population Mean - 1500

H0 - 1500

H1 - 1800

Step 2:- Setting default confidence interval to be 0.05

Step 3:- As we are aware of the formula:-

$Z = x - \mu /\sigma$

i.e. Z = 1800-1500/500 = 0.6

Step 4 :- Check for p value in Z table, thus we have value of 0.6 to be 0.7257, Thus probability amount of waste he receives is significantly more than the other tip would be (1-0.7257) = 0.2743

Step 5:- Now the p value is greater than significance value 0.05. Thus we cannot reject null hypothesis as its less than the significant value

d) Now as we know the p value to be 0.2743 lets compare the results with different significant value:-

Significance Vs Results

0.5	We can reject null hypothesis
1.0	We can reject null hypothesis
2.5	We can reject null hypothesis
5	We can reject null hypothesis
0.1	We cannot reject null hypothesis