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, 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 population of Blue Valley changed?

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

29 | 26 | 65 | 66 | 24 |

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

0.01

State the null and alternate hypotheses.

*H*_{0}: Time ten years ago and today are
independent. & *H*_{1}: Time ten years ago and
today are not independent.

*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}: 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.

*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.

(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.

Answer #1

The statistical software output for this problem is :

(i)

*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.

(ii)

sample test statistic = 11.11

(iii)

0.010 < *P*-value < 0.025

(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 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...

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 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...

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