Question

The table below contains information from a survey among 499 participants classified according to their age groups. The second column shows the percentage of obese people per age class among the study participants. The last column comes from a different study at the national level that shows the corresponding percentages of obese people in the same age classes in the USA. Perform a hypothesis test at the 5% significance level to determine whether the survey participants are a representative sample of the USA obese population.

Age Class (Years)	Obese (Percentage)	Expected USA average (Percentage)
20–30	75.0	32.6
31–40	25.5	32.6
41–50	13.6	36.6
51–60	21.9	36.6
61–70	20.0	39.7

• Part (a)

  State the null hypothesis.

  The distributions of surveyed obese and expected obese are the same.The distributions of surveyed obese and expected obese are not the same.    Surveyed obese fit the distribution of expected obese.Surveyed obese do not fit the distribution of expected obese.Surveyed obese and expected obese are independent events.

• Part (b)

  State the alternative hypothesis.

  The distributions of surveyed obese and expected obese are the same.The distributions of surveyed obese and expected obese are not the same.    Surveyed obese fit the distribution of expected obese.Surveyed obese do not fit the distribution of expected obese.Surveyed obese and expected obese are independent events.

• Part (c)

  What are the degrees of freedom? (Enter an exact number as an integer, fraction, or decimal.)
  (No Response)

• Part (d)

  State the distribution to use for the test.

  χ²₅

  χ²₄   

  t₅

  t₄

• Part (e)

  What is the test statistic? (Round your answer to two decimal places.)
  (No Response)

• Part (f)

  What is the p-value? (Round your answer to four decimal places.)
  (No Response)
  
  Explain what the p-value means for this problem.

  If H₀ is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.If H₀ is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.    If H₀ is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.If H₀ is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.

• Part (g)

  Sketch a picture of this situation. Shade the region(s) corresponding to the p-value.

• Part (h)

  Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write the appropriate conclusion.(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
  α = (No Response)
  
  (ii) Decision:

  reject the null hypothesisdo not reject the null hypothesis    

  
  (iii) Reason for decision:

  Since α > p-value, we do not reject the null hypothesis.Since α < p-value, we do not reject the null hypothesis.    Since α < p-value, we reject the null hypothesis.Since α > p-value, we reject the null hypothesis.

  
  (iv) Conclusion:

  There is sufficient evidence to conclude that the surveyed obese do not fit the distribution of expected obese.There is not sufficient evidence to conclude that the surveyed obese do not fit the distribution of expected obese.

Answer 1

(a)

Surveyed obese fit the distribution of expected obese.

(b)

Surveyed obese do not fit the distribution of expected obese.

(c)

There are 5 categories so degree of freedom is

df = 5 - 1 = 4

(d)

χ²₄

(e)

Following table shows the calculations for test statistics:

O	E	(O-E)^2/E
75	32.6	55.1460123
25.5	32.6	1.54631902
13.6	36.6	14.4535519
21.9	36.6	5.90409836
20	39.7	9.77556675
Total		86.8255483

The test statistics is

(f)

The p-value is: 0.0000

Excel function used for p-value: "=CHIDIST(86.83, 4)"

If H₀ is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.

(g)

Followig is the graph:

(h)

(i) α = 0.05

(ii)

reject the null hypothesis

(iii)

Since α > p-value, we reject the null hypothesis.

(iv)

There is sufficient evidence to conclude that the surveyed obese do not fit the distribution of expected obese