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a guidance counselor selected random samples of 25 ninth graders, 25 te…

Question

a guidance counselor selected random samples of 25 ninth graders, 25 tenth graders, 25 eleventh graders, and 25 twelfth graders from her large high school. she asked each student whether they have a job. the results are displayed in the tables.
observed counts:

expected counts:

she decides to test these hypotheses:
( h_0 ): there is no difference in the distribution of job status for all 9th, 10th, 11th, and 12th graders at this
for which categories were there more students observed than expected?
there were more 9th graders than expected who did not have a job, and there were more 10th, 11th, and 12th graders than expected who did have a job.
there were more 9th and 10th graders than expected who did not have a job, and there were more 11th and 12th graders than expected who did have a job.
there were more 9th, 10th, and 11th graders than expected who did not have a job, and there were more 12th graders than expected who did have a job.
there were more 11th and 12th graders than expected who did not have a job, and there were more 9th and 10th graders than expected who did have a job.

Explanation:

Brief Explanations

Compare the observed and expected counts. For 9th graders: Observed "No" (22) > Expected "No" (12.75). For 10th graders: Observed "No" (13) < Expected "No" (12.75) is false, actually Observed "No" (13) > Expected "No" (12.75) is incorrect (wait no, wait 12.75 vs 13: 13>12.75? No, 13 - 12.75=0.25, yes 13>12.75. Wait no, wait 12.75 is expected "No" for 10th. Observed "No" for 10th is 13. 13>12.75. But wait the first option: For 9th, observed "No" (22) > expected "No" (12.75). For 10th, observed "Yes" is 12, expected "Yes" is 12.25. 12<12.25. But wait no, the first option says "there were more 9th graders than expected who did not have a job (correct: 22>12.75), and more 10th,11th,12th than expected who did have a job. Let's check:

  • 10th: observed "Yes" =12, expected "Yes"=12.25. 12<12.25. No. Wait no, wait the first option's description: "there were more 9th graders than expected who did not have a job (22>12.75: correct). And for 10th: observed "Yes" (12) < expected "Yes" (12.25). But wait no, wait the first option says "more 10th... who did have a job". But 12<12.25. Wait no, wait maybe a miscalculation. Wait the observed counts:

For "Yes":
9th:3 (expected 12.25). 3<12.25.
10th:12 (expected12.25). 12<12.25.
11th:15 (expected12.25). 15>12.25.
12th:19 (expected12.25).19>12.25.
For "No":
9th:22 (expected12.75).22>12.75.
10th:13 (expected12.75).13>12.75.
11th:10 (expected12.75).10<12.75.
12th:6 (expected12.75).6<12.75.
So:

  • 9th "No": observed > expected.
  • 10th "No":13>12.75 (observed > expected). But the first option says "more 10th... who did have a job". No, for 10th "No" (job not had) is observed > expected. Wait the first option's wording: "there were more 9th graders than expected who did not have a job (correct: 22>12.75), and there were more 10th,11th,12th graders than expected who did have a job". But for 10th "Yes" (job had) is 12 <12.25 (expected). For 11th "Yes":15>12.25. For12th "Yes":19>12.25. But 10th "Yes" is less. Wait no, maybe a typo in options. Wait re - check:

The first option: "there were more 9th graders than expected who did not have a job (22>12.75: correct), and there were more 10th,11th,12th graders than expected who did have a job". But 10th "Yes" (job had) is 12 vs 12.25 (no). But wait maybe the options' first one:
Wait the observed "No" (job not had):
9th:22 (expected12.75) → observed > expected.
10th:13 (expected12.75) → observed > expected. But the first option says "more 10th... who did have a job". No. But wait maybe the options' first one is correct in another way. Wait the total in each grade is 25.
For 9th: Observed "No" (22) > expected "No" (12.75). So "more 9th... did not have a job".
For 10th: Observed "Yes" (12) < expected "Yes" (12.25). But observed "No" (13) > expected "No" (12.75). But the first option says "more 10th... who did have a job". No. But wait maybe a miscalculation. Wait the expected counts:
Total "Yes" =3 + 12+15+19=49. Total students=25×4=100. Expected "Yes" per grade:49/4 =12.25. Expected "No" per grade: (100 - 49)/4=12.75.
So:

  • 9th: Observed "No" (22) >12.75.
  • 10th: Observed "Yes" (12) <12.25; Observed "No" (13) >12.75.
  • 11th: Observed "Yes" (15) >12.25; Observed "No" (10) <12.75.
  • 12th: Observed "Yes" (19) >12.25; Observed "No" (6) <12.75.

So:

  • More 9th "No" (did not have job) than expected.
  • More 11th "Yes" (had job) than expected.
  • More 12th "Yes" (had job) than expected.
  • 10th: Observed "No" (13) >12.75 (did not have job more than expected). But the first option says "more 10th... who did have a job". No. But maybe the option's wording is wrong. Wait re - check the…

Answer:

There were more 9th and 10th graders than expected who did not have a job, and there were more 11th and 12th graders than expected who did have a job.