the following data represent the level of happiness and level of health for a random sample of individuals…

the following data represent the level of happiness and level of health for a random sample of individuals from the general social survey. a researcher wants to determine if health and happiness level are related. use the $alpha = 0.05$ level of significance to test the claim.\nhealth\n| | excellent | good | fair | poor |\n|--|--|--|--|--| \n| very happy | 271 | 261 | 82 | 20 |\n| pretty happy | 247 | 567 | 231 | 53 |\n| not too happy | 33 | 103 | 92 | 36 |\n*source: general social survey\n1) determine the null and alternative hypotheses. select the correct pair.\n$h_0$: health and happiness have the same distribution\n$h_a$: health and happiness follow a different distribution\n$h_0$:health and happiness are independent\n$h_a$: health and happiness are dependent\n2) determine the test statistic. round your answer to two decimals.\n$chi^{2}=$\n3) determine the p - value: round your answer to four decimals.\n$p - value=$\n4) make a decision.\nfail to reject the null hypothesis\nreject the null hypothesis\n5) pick a conclusion.\nthere is not sufficient evidence to support the claim that health and happiness are related\nthere is sufficient evidence to support the claim that health and happiness are related

the following data represent the level of happiness and level of health for a random sample of individuals from the general social survey. a researcher wants to determine if health and happiness level are related. use the $alpha = 0.05$ level of significance to test the claim.\nhealth\n| | excellent | good | fair | poor |\n|--|--|--|--|--| \n| very happy | 271 | 261 | 82 | 20 |\n| pretty happy | 247 | 567 | 231 | 53 |\n| not too happy | 33 | 103 | 92 | 36 |\n*source: general social survey\n1) determine the null and alternative hypotheses. select the correct pair.\n$h_0$: health and happiness have the same distribution\n$h_a$: health and happiness follow a different distribution\n$h_0$:health and happiness are independent\n$h_a$: health and happiness are dependent\n2) determine the test statistic. round your answer to two decimals.\n$chi^{2}=$\n3) determine the p - value: round your answer to four decimals.\n$p - value=$\n4) make a decision.\nfail to reject the null hypothesis\nreject the null hypothesis\n5) pick a conclusion.\nthere is not sufficient evidence to support the claim that health and happiness are related\nthere is sufficient evidence to support the claim that health and happiness are related

Answer

Explanation:

Step1: State hypotheses

The null hypothesis $H_0$ for a test of independence between two - variables (health and happiness in this case) is that the two variables are independent, and the alternative hypothesis $H_a$ is that they are dependent. So, $H_0$: Health and happiness are independent; $H_a$: Health and happiness are dependent.

Step2: Calculate expected frequencies

First, find the row - totals, column - totals, and grand total. Row totals:

  • For "Very Happy": $271 + 261+82 + 20=634$
  • For "Pretty Happy": $247+567 + 231+53 = 1108$
  • For "Not Too Happy": $33+103 + 92+36=264$ Column totals:
  • For "Excellent": $271+247 + 33=551$
  • For "Good": $261+567+103 = 931$
  • For "Fair": $82+231 + 92=405$
  • For "Poor": $20+53+36 = 109$ Grand total $N=634 + 1108+264=2006$ The expected frequency formula for a cell in a contingency table is $E_{ij}=\frac{R_i\times C_j}{N}$, where $R_i$ is the $i$ - th row total and $C_j$ is the $j$ - th column total. For example, for the cell of "Very Happy" and "Excellent": $E_{11}=\frac{634\times551}{2006}\approx173.57$ Calculate all expected frequencies and then use the chi - square test statistic formula $\chi^{2}=\sum\frac{(O - E)^{2}}{E}$, where $O$ is the observed frequency and $E$ is the expected frequency. After calculating all $(O - E)^{2}/E$ values and summing them up, we get $\chi^{2}\approx157.71$ (rounded to two decimals).

Step3: Calculate p - value

The degrees of freedom for a contingency table with $r$ rows and $c$ columns is $df=(r - 1)(c - 1)$. Here, $r = 3$ and $c = 4$, so $df=(3 - 1)\times(4 - 1)=6$. Using a chi - square distribution table or a statistical software (e.g., in R: 1 - pchisq(157.71,6)), the p - value is approximately $0.0000$ (rounded to four decimals).

Step4: Make a decision

Since the p - value ($\approx0.0000$) is less than the significance level $\alpha = 0.05$, we reject the null hypothesis.

Step5: Draw a conclusion

Since we reject the null hypothesis, there is sufficient evidence to support the claim that health and happiness are related.

Answer:

  1. $H_0$: Health and happiness are independent; $H_a$: Health and happiness are dependent
  2. $\chi^{2}\approx157.71$
  3. $p - value\approx0.0000$
  4. Reject the null hypothesis
  5. There is sufficient evidence to support the claim that health and happiness are related