Question

es The table below is based on data from a poll. Use a 0.05 significance level to test the claim that gender and left-handedness are independent. Left-handed Not Left-handed Women 178 115 Men 57 52 =cd ha Are gender and left-handedness independent? r. O A. No, there is sufficient evidence to warrant rejection of the claim of independence between nt gender and left-handedness. OB. Yes, there is sufficient evidence to warrant rejection of the claim of independence between gender and left-handedness. ta O C. Yes, there is not sufficient evidence to warrant rejection of the claim of independence between gender and left-handedness. th OD. No, there is not sufficient evidence to warrant rejection of the claim of independence between gender and left-handedness en en .th th th

Answer 1

Solution:

Given:

Left-handed Not Left-handed Total Women 178 115 293 Men 57 52 109 Total 235 167 402

Expected values can be found :

Expected Values Left-handed Not Left-handed Total Women 235 x 293 402 171.281 167 x 293 402 121.719 293 Men 235x 109 = 63.719 167 x 109 402 = 45.281 109 402 Total 235 167 402

We have to calculate the term,

$\frac{(E-O)^{2}}{E}$

Therefore,

Squared Distances Left-handed Not Left-handed Women (178-171.281 171.281 0.264 (115-121.719) 121.719 -0.371 Men (57-63.719) = 0.708 (52-45.2812 45.281 = 0.997 63.719

The following null and alternative hypotheses need to be tested:

H0​: The two variables are independent

Ha​: The two variables are dependent

This corresponds to a Chi-Square test of independence.

Based on the information provided, the significance level is α=0.05 , 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>3.841}.

The Chi-Squared statistic is computed as follows:

x = ☺ (0, - E.;)? = 0.264 +0.708 + 0.371 + 0.997 = 2.34 i=1

Since it is observed that χ2=2.34 ≤ χc2​=3.841, it is then concluded that the null hypothesis is not rejected.

Conclusion:

No, there is not sufficient evidence to warrant rejection of the claim of independence between gender and left-handedness (optionD)

Done