Are birthdays "evenly distributed" throughout the year, or are they more common during some parts of the year than others? Owners of a children's toy store chain asked this question. Some data collected by the chain are summarized in the table below. The data were obtained from a random sample of 180 people. The birthdate of each person was recorded, and each of these dates was placed into one of four categories: winter (December 21-March 20), spring (March 21-June 20), summer (June 21-September 20), and fall (September 21-December 20). The numbers in the first row of the table are the frequencies observed in the sample for these season categories. The numbers in the second row are the expected frequencies under the assumption that birthdays are equally likely during each season of the year. The bottom row of numbers gives the following value for each of the season categories. (Each expert I have asked has gotten the critical value wrong so I added a picture of the correct formula to use.)

Answer the following to summarize the test of the hypothesis that birthdays are equally likely during each season of the year. Use the 0.10 level of significance for the test.

A. Find the value of the test statistic. (Round your answer to two or more decimal places.)

B. Find the critical value. (Round your answer to two or more decimal places.)

C. Can we reject the hypothesis that birthdays are equally likely during each season of the year?
Transcribed Image Text:Observed frequency fo Expected frequency JE 2 (ƒo¯ƒE)² SE Part 2 Winter Spring Summer 42 45 .200 46 45.00 0.022 54 Fall 1.800 38 45.00 45] 1.089 Total 180 X Answer the following to summarize the test of the hypothesis that birthdays are equally likely during each season of the year. Use the 0.10 level of significance for the test.
Transcribed Image Text:(c) Finding the critical value 2 The value that cuts off an area of 0.10 in the right tail of a chi-square distribution with 4 degrees of freedom is about 7.78, that is, X0.10 7.78 for this distribution. Thus, the critical value for our test is 7.78. See the figure below. We consider only the right tail probability because only large values of the test statistic will lead us to reject the null hypothesis. Rejection region for a - 0.10 7.78

Question

Are birthdays &#34;evenly distributed&#34; throughout the year, or are they more common during some parts of the year than others? Owners of a children's toy store chain asked this question. Some data collected by the chain are summarized in the table below. The data were obtained from a random sample of 180 people. The birthdate of each person was recorded, and each of these dates was placed into one of four categories: winter (December 21-March 20), spring (March 21-June 20), summer (June 21-September 20), and fall (September 21-December 20). The numbers in the first row of the table are the frequencies observed in the sample for these season categories. The numbers in the second row are the expected frequencies under the assumption that birthdays are equally likely during each season of the year. The bottom row of numbers gives the following value for each of the season categories. (Each expert I have asked has gotten the critical value wrong so I added a picture of the correct formula to use.) Answer the following to summarize the test of the hypothesis that birthdays are equally likely during each season of the year. Use the 0.10 level of significance for the test.A. Find the value of the test statistic. (Round your answer to two or more decimal places.) B. Find the critical value. (Round your answer to two or more decimal places.)C. Can we reject the hypothesis that birthdays are equally likely during each season of the year? ObservedfrequencyfoExpectedfrequencyJE2(ƒo¯ƒE)²SEPart 2Winter Spring Summer4245.2004645.000.02254Fall1.8003845.00 45]1.089Total180XAnswer the following to summarize the test of the hypothesis that birthdays are equally likely during each season of the year. Use the 0.10 level ofsignificance for the test. (c) Finding the critical value2The value that cuts off an area of 0.10 in the right tail of a chi-square distribution with 4 degrees of freedom is about 7.78, that is, X0.107.78for this distribution. Thus, the critical value for our test is 7.78. See the figure below. We consider only the right tail probability because only largevalues of the test statistic will lead us to reject the null hypothesis.Rejection region for a - 0.107.78

Accepted Answer

fo
fe
(fo-fe)2/fe

Winter
42
45
0.2

Spring
46
45
0.022222

Summer
54
45
1.8

Fall
38
45…