Question

A sociologist is studying the age of the population in Blue Valley. Ten years ago, the population was such that 19% were under 20 years old, 12% were in the 20- to 35-year-old bracket, 34% were between 36 and 50, 23% were between 51 and 65, and 12% were over 65. A study done this year used a random sample of 210 residents. This sample is given below. At the 0.01 level of significance, has the age distribution of the population of Blue Valley changed?

Under 20	20 - 35	36 - 50	51 - 65	Over 65
28	25	68	65	24

(i) Give the value of the level of significance.


State the null and alternate hypotheses.

H₀: The population 10 years ago and the population today are independent.
H₁: The population 10 years ago and the population today are not independent. H₀: Ages under 20 years old, 20- to 35-year-old, between 36 and 50, between 51 and 65, and over 65 are independent.
H₁: Ages under 20 years old, 20- to 35-year-old, between 36 and 50, between 51 and 65, and over 65 are not independent.     H₀: Time ten years ago and today are independent.
H₁: Time ten years ago and today are not independent. H₀: The distributions for the population 10 years ago and the population today are the same.
H₁: The distributions for the population 10 years ago and the population today are different.

(ii) Find the sample test statistic. (Round your answer to two decimal places.)


(iii) Find or estimate the P-value of the sample test statistic.

P-value > 0.100 0.050 < P-value < 0.100     0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 P-value < 0.005

(iv) Conclude the test.

Since the P-value < α, we do not reject the null hypothesis. Since the P-value < α, we reject the null hypothesis.     Since the P-value ≥ α, we do not reject the null hypothesis. Since the P-value ≥ α, we reject the null hypothesis.

(v) Interpret the conclusion in the context of the application.

At the 1% level of significance, there is insufficient evidence to claim that the age distribution of the population of Blue Valley has changed. At the 1% level of significance, there is sufficient evidence to claim that the age distribution of the population of Blue Valley has changed.    

Answer 1

Ans:

Level of significance=0.01

H₀: The distributions for the population 10 years ago and the population today are the same.
H₁: The distributions for the population 10 years ago and the population today are different.

	pi	fo	fe	(fo-fe)^2/fe
under 20	19%	28	39.9	3.55
20-35	12%	25	25.2	0.00
36-50	34%	68	71.4	0.16
51-65	23%	65	48.3	5.77
over 65	12%	24	25.2	0.06
Total	100%	210	210	9.54

Test statistic:

Chi square=9.54

df=5-1=4

p-value=CHIDIST(9.54,4)=0.0489

0.025 < P-value < 0.050

Since the P-value ≥ α, we do not reject the null hypothesis.

At the 1% level of significance, there is insufficient evidence to claim that the age distribution of the population of Blue Valley has changed.