Question

A researcher whether introverts are more likely to prefer online classes than extraverts. They decide to test this theory by asking a sample of n = 80 people (a) whether they are an introvert or an extravert, and (b) whether they prefer online or InPerson classes.

Below is the data from the study:

	Online	InPerson
Introvert	ƒo = 8	ƒo = 19
Extravert	ƒo = 23	ƒo = 30

(a) Compute df and determine the critical value for χ² for a hypothesis test with α = 0.01.

df =
Critical value of χ² =

(b) Determine the expected frequencies (ƒ_e) for each of the cells in the table: (Use 3 decimals)

	Online	No Online
Introvert	ƒe =	ƒe =
No Introvert	ƒe =	ƒe =

(c) Calculate the chi-square statistic (χ²): (Use 3 decimals)

χ² =

(d) Write the results as you would write it for the research literature.

Answer 1

	Online	No Online	Total
Introvert	8	19	27
No Introvert	23	30	53
Total	31	49	80

The expected values are computed in terms of row and column totals. In fact, the formula is

, where R_i corresponds to the total sum of elements in row i, C_j​ corresponds to the total sum of elements in column j, and T is the grand total.

The table below shows the calculations to obtain the table with expected values:

Expected Values	Online	No Online	Total
Introvert			27
No Introvert			53
Total	31	49	80

Based on the observed and expected values, the squared distances can be computed according to the following formula: (E - O)^2/E.

The table with squared distances is shown below:

Squared Distances	Online	No Online
Introvert
No Introvert

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H_0​: The two variables are independent

H_a: The two variables are dependent

This corresponds to a Chi-Square test of independence.

Rejection Region

Based on the information provided, the significance level is α=0.01 ,

the number of degrees of freedom is df = (2−1)×(2−1)=1,

so then the rejection region for this test is

R={χ2:χ2>6.635}.

Test Statistics

The Chi-Squared statistic is computed as follows:

Decision about the null hypothesis

Since it is observed that χ2=1.428≤χc2​=6.635,

it is then concluded that the null hypothesis is not rejected.

Conclusion

It is concluded that the null hypothesis Ho is not rejected.

Therefore, there is NOT enough evidence to claim that the two variables are dependent, at the 0.01 significance level.

The corresponding p-value for the test is

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