Question

22.HE.A: It has been assumed that children are born with equal frequency with respect to days of the week. Randomly selected birth records are obtained and results are listed in the table below.

Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
36	55	62	60	60	58	48

1. State the test that should be used.
2. Write the null and alternative hypotheses.
3. Check the three conditions.
4. How many degrees of freedom should we use in the χ² test?
5. Compute the χ² statistic and p-value.
6. Which cell contributed the most to the χ²? Which one contributed the least?
7. State the conclusion in context.

Answer 1

a)

Chi Square test of goodness of fit

b)

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested Ha: Some of the population proportions differ from the values stated in the null hypothesis

c)

The following conditions are all satisfied:

• The sampling method used is simple random sampling.
• The variable under study - Day of the week is categorical.
• The expected value of the number of sample observations in each level of the variable is at least 5.

d)

Based on the information provided, the significance level is α=0.05, the number of degrees of freedom is df=7−1=6

e)

The following table is obtained (fo-fe)2lfe (36-54.143)/54.143 6.08 (55-54.143)/54.1430.014 (62-54.143)/54.143-1.14 (60-54.143)/54.1430.634 (60-54.143)/54.1430.634 (58-54.143)/54.1430.275 (48-54.143)/54.1430.697 Categories Sunday Monday Tuesday Wednesday Thursday Friday Saturday Observed Expected 379 1/7-54.143 379 1/7-54.143 379 1/7-54.143 379 1/7-54.143 379 1/7-54.143 379 1/7-54.143 379 1/7-54.143 36 62 60 60 58 379 Sum - 379 9.472

Test Statistics

The Chi-Squared statistic is computed as follows:

O- E) Ei 6.08+0.014+1.14+0.634+0.634 +0.275+0.697 9.472

f)

Maximum - Sunday

Minimum - Monday

g)

At 5% level of significance, since it is observed that χ2=9.472≤χc2​=12.592, it is then concluded that the null hypothesis is not rejected.Therefore, there is NOT enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the \α=0.05 significance level.

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