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, 11% 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 |
|---|---|---|---|---|
| 28 | 25 | 66 | 66 | 25 |
(i) Give the value of the level of significance.
State the null and alternate hypotheses.
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: 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 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 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.
i)
value of the 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 goodness of fit test:
| observed | Expected | Chi square | |||
| category | Probability(p) | Oi | Ei=total*p | R2i=(Oi-Ei)2/Ei | |
| <20 | 0.200 | 28.000 | 42.000 | 4.667 | |
| 20-35 | 0.110 | 25.000 | 23.100 | 0.156 | |
| 36-50 | 0.340 | 66.000 | 71.400 | 0.408 | |
| 51-65 | 0.240 | 66.000 | 50.400 | 4.829 | |
| >65 | 0.110 | 25.000 | 23.100 | 0.156 | |
| total | 1.000 | 210 | 210 | 10.216 |
sample test statistic =10.22
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.
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 20% were under 20 years old, 14% were in the 20- to 35-year-old bracket, 32% 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...
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