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, 12% were in the 20- to 35-year-old bracket, 34% were between 36 and 50, 24% were between 51 and 65, and 10% 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	29	66	67	19

(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

(i)

Level of significance = = 0.01

Correct option:

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

H1: The distribution forpopulation 10 years ago and the population today are different.

(ii)

Observed frequencies:

Under 20	20-35	36-50	51-65	Over 65	Total
29	29	66	67	19	210

Expected Frequencies:

Under 20	20-35	36-50	51-65	Over 65	Total
210X20/100=42	210X12/100=25.2	210X34/100=71.4	210X24/100=50.4	210X10/100=21	210

O          E                   (O - E)²/E

29         42                     4.02

29        25.2                  0.57

66          71.4               0.41

67           50.4             5.47

19             21                0.19

-----------------------------------------------

                          = 10.66

(iii)

= 10.66

ndf = 5- 1 = 4

From Technology, p-value = 0.0307

So,

Correct option:

0.025 < P-value < 0.050

(iv)

Since P-value = 0.0307 is greater than = 0.01, Fail to reject null hypothesis.

Correct option:

Since the P-value , we do not rejct the null hypothesis.

(v)

Correct option:

At the 1% level of significance, thereis insufficient evidence to claim that the age distribution of the population of Blue Valley haschanged.