Question

In: Statistics and Probability

A sociologist is studying the age of the population in Blue Valley. Ten years ago, the...

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.

Solutions

Expert Solution

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.


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