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, 24% were between 51 and 65, and 11% 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
27	29	66	65	23

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


State the null and alternate hypotheses.

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.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.

(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 > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-value < 0.005

(iv) Conclude the test.

Since the P-value < α, we 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 do not 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

The statistical software output for this problem is :

(i)

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

(ii)

sample test statistic = 9.38

(iii)

0.050 < P-value < 0.100

(iv)

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

(v)

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