Question

In: Math

Data from adoption studies on handedness indicate that the effects of shared biological heritage are more...

Data from adoption studies on handedness indicate that the effects of shared biological heritage are more powerful determinants of hand preference than sociocultural factors. Biological offspring were found to show nonrandom distributions of right and non-right handedness as a function of parental handedness. In contrast, the handedness distribution of adopted children as a function of parental handedness was essentially random. The following table gives the handedness of biological offspring as a function of parental handedness.
Biological Offspring
Parental Handedness
(father × mother)
Right-Handed Left-Handed
Right × Right 298 37
Right × Left 29 6
Left × Right 24 6
(a) Find the values in cells (2,2) and (3,2) of the expected table.
(b) Can a chi-square analysis be performed on the above table?
(c) Combine the last two rows in the above table to create a new 2 × 2 table. (The meaning of the 2nd row in this new table would be "at least one of the parents is left-handed".)
If we use the resulting 2 × 2 table to test the hypothesis that the handedness of the biological offspring is independent of the handedness of the parents using the 1% significance level, what is the value of the test statistic?
(d) Find the critical value for the test in (c).
(e) What is the conclusion?

Problem #2(a):

expected values (correct to 3 decimals)
separate your answers with a comma

(A) Yes, because at least one of the expected frequencies is less than 5. (B) Yes, because at least one of the observed frequencies is less than 5. (C) No, because at least one of the expected frequencies is less than 5. (D) No, because the population is not normal. (E) Yes, because only some of the expected frequencies are less than 5. (F) No, because at least one of the observed frequencies is less than 5. (G) Yes, because all of the observed frequencies are at least 5.

Solutions

Expert Solution

Parental Handedness Right-Handed Left-Handed Total
Right × Right 298 37 335
Right × Left 29 6 35
Left × Right 24 6 30
Total 351 49 400

a) Expected value for (2, 2) = 49 * 35 / 400 = 4.288

Expected value for (2, 3) = 49 * 30 / 400 = 3.675

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b) Answer: (C) No, because at least one of the expected frequencies is less than 5.

---

c)

Parental Handedness Right-Handed Left-Handed Total
Right × Right 298 37 335
at least one is left-handed 53 12 65
Total 351 49 400
Expected Frequencies
Right-Handed Left-Handed Total
Right × Right 351 * 335 / 400 = 293.9625 49 * 335 / 400 = 41.0375 335
at least one is left-handed 351 * 65 / 400 = 57.0375 49 * 65 / 400 = 7.9625 65
Total 351 49 400
(fo-fe)²/fe
Right × Right (298 - 293.9625)²/293.9625 = 0.0555 (37 - 41.0375)²/41.0375 = 0.3972
Right × Left (53 - 57.0375)²/57.0375 = 0.2858 (12 - 7.9625)²/7.9625 = 2.0473

Test statistic:

χ² = ∑ ((fo-fe)²/fe) = 2.7858

---

d) df = (r-1)(c-1) = 1

Critical value:

χ²α = CHISQ.INV.RT(0.01, 1) = 6.6349

---

e) Decision:

Do not reject the null hypothesis.


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