Question

Annual inflation in Canada has been low for several years. Suppose a provincial survey is conducted by a polling organization, and Saskatchewanians answered the question: “are you not at all concerned, somewhat concerned, or very concerned that inflation will climb?”. The results are given below.

Not at all concerned	Somewhat concerned	Very concerned
120	360	230

The same survey was conducted in Ontario, and it was found that 20% of Ontarians were very concerned, 45% were somewhat concerned, and 35% were not at all concerned. The polling organization is interested in determining if the proportions of people who are very concerned, somewhat concerned, or not at all concerned about inflation rising are the same in Saskatchewan as they are in Ontario. Conduct an appropriate hypothesis test by filling in the blanks below.

1) State the null and alternative hypotheses [1 mark]:

2) Calculate the test statistic. Enter your value below. Note: you may use Excel to calculate the test statistic, and it doesn’t matter if you round for each individual calculation or just at the end, you’ll get the same value. No need to show your work.
[1 mark]

3) The critical value is from a chi-squared distribution on 2 degrees of freedom, which is 5.99147.

4) Based on this critical value, do you reject the null hypothesis (Yes or No)? [1 mark]

Answer 1

Observed (Oi )	Expected ( Ei)	Oi-Ei	(Oi-Ei)^2	(Oi-Ei)^2/Ei
0.169	0.35	-0.18	0.0328	0.0937
0.507	0.45	0.057	0.0032	0.0071
0.3239	0.2	0.124	0.0154	0.077

observed freq are Oi
not at all concerned = 120/((120+360+230)) = 0.169
somewhat concerned = 360/((120+360+230)) = 0.507
very concerned= 230/((120+360+230)) = 0.324
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expected frequencies are Ei
20% were very concerned
45% were somewhat concerned
35% were not at all concerned   
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and claiming hypothesis is
null, Ho: no association exists b/w the proportions of people who are very concerned
alternative, H1: association exists b/w the proportions of people who are very concerned
level of significance, alpha = 0.05
from standard normal table, chi square value at right tailed, ᴪ^2 alpha/2 =5.9915
since our test is right tailed,reject Ho when ᴪ^2 o > 5.9915
we use test statistic ᴪ^2 o = Σ(Oi-Ei)^2/Ei
from the table take sum of (Oi-Ei)^2/Ei, we get ᴪ^2 o = 0.1778
critical value
the value of |ᴪ^2 alpha| at los 0.05 with d.f, n - 1 = 3 - 1 = 2 is 5.9915
we got | ᴪ^2| =0.1778 & | ᴪ^2 alpha | =5.9915
make decision
hence value of | ᴪ^2 o | < | ᴪ^2 alpha | and here we do not reject Ho
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null, Ho: no association exists b/w the proportions of people who are very concerned
alternative, H1: exists association exists b/w the proportions of people who are very concerned
test statistic: 0.1778
critical value: 5.9915
decision: do not reject Ho