Question

Statistics Canada conducts monthly surveys to provide the public information on key labour market variables. For the current month, Stats Canada surveys 100 residents in Quebec City and 400 residents in Montreal. The reported variance for the number of unemployed in Quebec City is 90, while that for Montreal is 160, The historical variance for the number of unemployed for both cities is 100.

a) Find whether the variability in the number of unemployed differs between the historical value and the one reported in the sample for the city of Montreal at the 5% level of significance.

b) Find whether the variability in the number of unemployed differs across the two cities at the 5% level of significance.

Answer 1

a)

Given,

The historical variance for the number of unemployed for both cities is 100

Hypothesied Variance :

Stats Canada surveys 400 residents in Montreal

Sample size : n = 400

The reported variance for the number of unemployed in Montreal is 160

Sample Variance : s² = 160

Null hypothesis : H_o : There is no difference in variability in the number of unemployed between the historical value and the one reported in the sample for the city of Montreal

Null hypothesis :

Alternate hypothesis : H_a : The variability in the number of unemployed differs between the historical value and the one reported in the sample for the city of Montreal

Alternate hypothesis :

Given
Hypothesied Variance:	100
Sample Variance : s2	160
Sample Size : n	100
Level of significance :	5%(0.05)
Degrees of Freedom : n-1 =100-1=99	99

As P-Value i.e. is greater than Level of significance i.e (P-value:2 > 0.05:Level of significance); Fail to Reject Null Hypothesis

Sufficient evidence to concclude that the variability in the number of unemployed differs between the historical value and the one reported in the sample for the city of Montreal at the 5% level of significance

CHISQ.DIST.RT function

Returns the right-tailed probability of the chi-squared distribution.

The χ2 distribution is associated with a χ2 test. Use the χ2 test to compare observed and expected values. For example, a genetic experiment might hypothesize that the next generation of plants will exhibit a certain set of colors. By comparing the observed results with the expected ones, you can decide whether your original hypothesis is valid.

Excel function CHISQ.DIST.RT(x,deg_freedom) is being used to obtain p-value.

Syntax

CHISQ.DIST.RT(x,deg_freedom)

The CHISQ.DIST.RT function syntax has the following arguments:

• X     Required. The value at which you want to evaluate the distribution.

• Deg_freedom     Required. The number of degrees of freedom.

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b)

Stats Canada surveys 400 residents in Montreal

Sample size for Montreal: n₁ = 400

The reported variance for the number of unemployed in Montreal is 160

Sample Variance for Montreal : s₁² = 160

Stats Canada surveys 100 residents in Quebec City

Sample size for Qubec city : n₂ = 100

The reported variance for the number of unemployed in Quebec City is 90

Sample variance for Qubec city : s₂² = 90

Null Hypothesis : H_o :  there is difference in the variability in the number of unemployed across the two cities

Alternate Hypothesis :H_a : the variability in the number of unemployed differs across the two cities

Given
n_{1} : Sample Size of Sample 1	400
n_{2} : Sample Size of Sample 2	100
s_{1} : Sample Variance of Sample 1	160
s_{2} : Sample Variance of Sample 2:	90
Degrees of Freedom : Numerator : n_{1} -1;400-1=399	399
Degrees of Freedom : Denominator : n_{2} -1;100-1=99	99
Level of Significance :	5%

As P-Value i.e. is less than Level of significance i.e (P-value:0.0008 < 0.05:Level of significance); Reject Null Hypothesis

Sufficent evidence to conlcude that the variability in the number of unemployed differs across the two cities at the 5% level of significance.

p-value is computed using Excel function

F.DIST.RT function

This article describes the formula syntax and usage of the F.DIST.RT function in Microsoft Excel.

Returns the (right-tailed) F probability distribution (degree of diversity) for two data sets. You can use this function to determine whether two data sets have different degrees of diversity. For example, you can examine the test scores of men and women entering high school and determine if the variability in the females is different from that found in the males.

Syntax

F.DIST.RT(x,deg_freedom1,deg_freedom2)

The F.DIST.RT function syntax has the following arguments:

• X     Required. The value at which to evaluate the function.

• Deg_freedom1     Required. The numerator degrees of freedom.

• Deg_freedom2     Required. The denominator degrees of freedom.