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, 13% were in the 20- to 35-year-old bracket, 29% were between 36 and 50, 24% were between 51 and 65, and 15% 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	27	65	65	26

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


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: 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.

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 > 0.100

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 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

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)

Applying chi square test:

	relative	observed	Expected	residual	Chi square
category	frequency	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
<20	0.190	27	39.90	-2.04	4.171
20-35	0.130	27	27.30	-0.06	0.003
36-50	0.290	65	60.90	0.53	0.276
51-65	0.240	65	50.40	2.06	4.229
>65	0.150	26	31.50	-0.98	0.960
total	1.000	210	210		9.640

sample test statistic X² =9.640

iii)

0.025 < P-value < 0.050

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.