Question

Based on sales over a six-month period, the five top-selling compact cars are Chevy Cruze, Ford Focus, Hyundai Elantra, Honda Civic, and Toyota Corolla.† Based on total sales, the market shares for these five compact cars were Chevy Cruze 24%, Ford Focus 21%, Hyundai Elantra 20%, Honda Civic 18%, and Toyota Corolla 17%. Suppose a sample of 400 compact car sales in one city showed the following number of vehicles sold.
Chevy Cruze 108
Ford Focus 92
Hyundai Elantra 64
Honda Civic 84
Toyota Corolla 52
Use a goodness of fit test to determine if the sample data indicate that the market shares for the five compact cars in Chicago are different than the market shares reported by Motor Trend. Using a .05 level of significance

The p-value is

What is your conclusion?

What market share differences, if any, exist in Chicago?

Answer 1

Answer:

Step-by-step explanation:

Hello!

The Chi-Square Goodness of Fit analysis objective is to test if the observations of a categorical variable follow the historical or theoretical values of the study population.

We know the population's sale proportions (reported by Motor trend) of five top-selling compact cars and need to compare it to a sample of 400 compact car sales of Chicago to know if these proportions are different.

Chevy Cruze 24%⇒ P(CC)=0.24

Ford Focus 21%⇒ P(FF)=0.21

Hyundai Elantra 20%⇒ P(HE)= 0.20

Honda Civic 18%⇒ P(HC)= 0.18

Toyota Corolla 17%⇒ P(TC)= 0.17

The study variable is X: top-selling compact cars in Chicago, categorized Chevy Cruze, Ford Focus, Hyundai Elantra, Honda Civic, Toyota Corolla.

Then the hypotheses are:

H₀: P(CC)=0.24; P(FF)=0.21; P(HE)= 0.20; P(HC)= 0.18; P(TC)= 0.17

H₁: There is a difference within the expected and the observed data.

α: 0.05

The statistic to use is:

$X^2=sum(\frac{(O_i-E_i)^2}{E_i} )~X^2_{k-1}$

Where

Oi is the observed frequency of each i-category

Ei is the expected frequency of each i-category

k is the number of categories.

The first step to obtaining the value of the statistic is to calculate the expected frequencies:

Ei= n * Pi

Where Pi is the theoretical proportion for the i-category stated in the null hypothesis.

$E_{CC}= 400*0.24= 96$

$E_{FF}= 400*0.21= 84$

$E_{HE}= 400*0.20= 80$

$E_{HC}= 400*0.18= 72$

$E_{TC}= 400*0.17= 68$

$X^2_{H_0}= (\frac{(108-96)^2}{96} )+(\frac{(92-84)^2}{84} )+(\frac{(64-80)^2}{80} )+(\frac{(84-72)^2}{72} )+(\frac{(52-68)^2}{68} )= 11.2266$

This test is always one-tailed to the right and so is the p-value, so you can calculate it as:

P(X²₄≥11.23)= 1 - P(X²₄<11.23)= 1 - 0.98= 0.02

The p-value is less than α, so the decision is to reject the null hypothesis. In other words, the market shares for the five compact cars in Chicago don't follow the market shares reported by Motor Trend.

I hope it helps!