Question

A sociologist is studying the age of the population in Blue Valley. Ten years ago, the population was such that 20% were under 20 years old, 10% were in the 20- to 35-year-old bracket, 32% were between 36 and 50, 24% were between 51 and 65, and 14% 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
29	26	65	66	24

(i) Give the value of the level of significance.
0.01

State the null and alternate hypotheses.

H₀: Time ten years ago and today are independent. & H₁: Time ten years ago and today are not independent.

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: 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

The statistical software output for this problem is :

(i)

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 = 11.11

(iii)

0.010 < P-value < 0.025

(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.