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, 14% were in the 20- to 35-year-old bracket, 34% were between 36 and 50, 24% were between 51 and 65, and 9% 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 |
---|---|---|---|---|

26 | 28 | 67 | 65 | 24 |

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

State the null and alternate hypotheses.

*H*_{0}: The population 10 years ago and the
population today are independent.

*H*_{1}: The population 10 years ago and the
population today are not independent.*H*_{0}: Time
ten years ago and today are independent.

*H*_{1}: Time ten years ago and today are not
independent. *H*_{0}: The
distributions for the population 10 years ago and the population
today are the same.

*H*_{1}: The distributions for the population 10
years ago and the population today are
different.*H*_{0}: Ages under 20 years old, 20- to
35-year-old, between 36 and 50, between 51 and 65, and over 65 are
independent.

*H*_{1}: Ages under 20 years old, 20- to
35-year-old, between 36 and 50, between 51 and 65, and over 65 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 < 0.010*P*-value < 0.005

(iv) Conclude the test.

Since the *P*-value ≥ *α*, we reject the null
hypothesis.Since the *P*-value ≥ *α*, we do not
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

Ans:

i) level of significance=0.01

*H*_{0}: The distributions for the population 10
years ago and the population today are the same.

*H*_{1}: The distributions for the population 10
years ago and the population today are different.

Under 20 | 20 - 35 | 36 - 50 | 51 - 65 | Over 65 | Total | |

Observed(fo) | 26 | 28 | 67 | 65 | 24 | 210 |

pi | 0.19 | 0.14 | 0.34 | 0.24 | 0.09 | 1 |

Expected(fe) | 39.9 | 29.4 | 71.4 | 50.4 | 18.9 | 210 |

(fo-fe)^2/fe | 4.84 | 0.07 | 0.27 | 4.23 | 1.38 |
10.79 |

ii)

sample test statistic:

Chi square=10.79

df=5-1=4

iii)

p-value=CHIDIST(10.79,4)=**0.0290**

0.025 < *P*-value < 0.050

iv)Since the ** P-value ≥ α,** we

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 population was such that 20% were under
20 years old, 13%were in the 20- to 35-year-old bracket, 30% were
between 36 and 50, 24% were between 51 and 65, and 13% 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...

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, 12% were in the 20- to 35-year-old bracket, 31% were
between 36 and 50, 23% 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...

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, 13% were in the 20- to 35-year-old bracket, 30% were
between 36 and 50, 24% were between 51 and 65, and 13% 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...

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, 12% 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...

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, 14% were in the 20- to 35-year-old bracket, 31% were
between 36 and 50, 24% were between 51 and 65, and 12% 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...

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...

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, 12% were in the 20- to 35-year-old bracket, 34% were
between 36 and 50, 23% were between 51 and 65, and 12% 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...

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, 14% were in the 20- to 35-year-old bracket, 29% 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...

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, 32% were
between 36 and 50, 24% were between 51 and 65, and 12% 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...

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, 12% were in the 20- to 35-year-old bracket, 32% were
between 36 and 50, 24% were between 51 and 65, and 13% 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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