Question

The age distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of a village are shown below.

Age (years) |
Percent of Canadian Population |
Observed Numberin the Village |

Under 5 | 7.2% | 44 |

5 to 14 | 13.6% | 80 |

15 to 64 | 67.1% | 289 |

65 and older | 12.1% | 42 |

Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

(a) What is the level of significance?

State the null and alternate hypotheses.

*H*_{0}: The distributions are different.

*H*_{1}: The distributions are
different.*H*_{0}: The distributions are the
same.

*H*_{1}: The distributions are
different. *H*_{0}: The
distributions are different.

*H*_{1}: The distributions are the
same.*H*_{0}: The distributions are the same.

*H*_{1}: The distributions are the same.

(b) Find the value of the chi-square statistic for the sample.
(Round your answer to three decimal places.)

Are all the expected frequencies greater than 5?

YesNo

What sampling distribution will you use?

uniformStudent's
*t* normalbinomialchi-square

What are the degrees of freedom?

(c) 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

(d) Based on your answers in parts (a) to (c), will you reject or
fail to reject the null hypothesis that the population fits the
specified distribution of categories?

Since the *P*-value > *α*, we fail to reject
the null hypothesis.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 fail to reject the null
hypothesis.

(e) Interpret your conclusion in the context of the
application.

At the 5% level of significance, the evidence is insufficient to conclude that the village population does not fit the general Canadian population.At the 5% level of significance, the evidence is sufficient to conclude that the village population does not fit the general Canadian population.

Answer #1

Solution:

**The Level of Significance is α=0.05**

The Null and Alternative Hypotheses are as follows:

**H0: The distributions are the
same.
vs. H1: The distributions are different.**

- The above test is the Test for Goodness of Fit where we need to check whether the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

Let the population have k mutually
exclusive classes and let us assume that according to the
hypothesis, the population proportion of the ith class is
p_{i.} If the frequency of the ith class in random samples
of size n from this population be f_{i,} we have the
following test statistic for the above hypothesis (Part (a))

Chi-squared = ∑ (f_{i,} - n
p_{i})^{2}/ n p_{i.} = ∑
(f_{i,}^{2}/ n p_{i.}) – n …..(1)

The above Test statistic ~
^{Chi-squared} with (k-1) degrees of freedom

For Computation we use the following table:

Class |
Observed freq (fi) |
% of Canadian Population |
Expected freq (npi) |
fi^2/npi |

< 5 |
44 |
7.2% |
7.2/100*455=32.76 |
44^2/32.76=59.0965 |

5 - 14 |
80 |
13.6% |
13.6/100*455=61.88 |
80^2/61.88=103.4260 |

15 - 64 |
289 |
67.1% |
67.1/100*455=305.305 |
289^2/305.305=273.5658 |

> 65 |
42 |
12.1% |
12.1/100*455=55.055 |
42^2/55.055=32.04068 |

Total |
455 |
100% |
455 |
468.1289 |

So the using the formula (1) we have,

**Chi-square****= ∑ (f**_{i,}^{2}/ n p_{i.}) – n =468.1289-455 (From the above table)

**
=13.129
(Round to three decimal places)**

**From the above table we can see all the expected frequencies are greater than 5. Thus it meets the assumption of using a chi-squared test.****We will use the Chi-squared test****The degrees of freedom=k-1, where k is the number of classes, Here k=4**

**So degrees of freedom
=4-1=3.**

**The p-value for the corresponding chi-squared test statistic =13.129 with degrees of freedom 3 and level of significance 0.05 is**

**=.004366**

**So, P-value <
0.005**

**Now at level of significance 0.05 we have,**

**P-value <α**

**So, we have - > Since
the** P**-value ≤** α**, we reject the
null hypothesis.**

**So, we can have the following conclusion:**

**At the 5% level of significance, the evidence is
sufficient to conclude that the village population does not fit the
general Canadian population. **

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
50
5 to 14
13.6%
78
15 to 64
67.1%
282
65 and older
12.1%
45
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
45
5 to 14
13.6%
71
15 to 64
67.1%
298
65 and older
12.1%
41
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
46
5 to 14
13.6%
80
15 to 64
67.1%
283
65 and older
12.1%
46
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age
(years)
Percent of
Canadian Population
Observed
Number
in the Village
Under 5
7.2%
45
5 to 14
13.6%
74
15 to 64
67.1%
286
65 and older
12.1%
50
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
45
5 to 14
13.6%
82
15 to 64
67.1%
288
65 and older
12.1%
40
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
50
5 to 14
13.6%
74
15 to 64
67.1%
282
65 and older
12.1%
49
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
46
5 to 14
13.6%
70
15 to 64
67.1%
298
65 and older
12.1%
41
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
51
5 to 14
13.6%
69
15 to 64
67.1%
292
65 and older
12.1%
43
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
52
5 to 14
13.6%
80
15 to 64
67.1%
280
65 and older
12.1%
43
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

The age distribution of the Canadian population and the age
distribution of a random sample of 455 residents in the Indian
community of a village are shown below.
Age (years)
Percent of Canadian Population
Observed Number
in the Village
Under 5
7.2%
50
5 to 14
13.6%
71
15 to 64
67.1%
289
65 and older
12.1%
45
Use a 5% level of significance to test the claim that the age
distribution of the general Canadian population fits the age...

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