Question

Data from the Motor Vehicle Department indicate that 80% of all licensed drivers are older than age 25.

a. In a sample of n=50 people who recently received speeding tickets, 33 were older than 25 years and the other 17 were age 25 or younger. Is the age distribution for this sample significantly different from the distribution for the population of licensed drivers? Use α= 0.05.

b. In a sample of n=50 people who recently received parking tickets, 36 were older than 25 years and the other 14 were age 25 or younger. Is the age distribution for this sample significantly different from the distribution for the population of licensed drivers? Use α= 0.05.

Answer 1

Part a

Here, we have to use z test for population proportion.

H₀: p = 0.80 versus H_a: p ≠ 0.80

This is a two tailed test.

We are given α = 0.05

Test statistic formula is given as below:

Z = (p̂ - p)/sqrt(pq/n)

We are given x = 33, n = 50

p̂ = x/n = 33/50 = 0.66

p = 0.80,

q = 1 – p = 1 – 0.80 = 0.20

Z = (0.66 – 0.80)/sqrt(0.80*0.20/50)

Z = -2.4749

P-value = 0.0133

(by using z-table)

P-value < α = 0.05

So, we reject the null hypothesis

At the 0.05 level of significance, there is sufficient evidence to conclude that age distribution for this sample is significantly different from the distribution for the population of licensed drivers.

Part b

Here, we have to use z test for population proportion.

H₀: p = 0.80 versus H_a: p ≠ 0.80

This is a two tailed test.

We are given α = 0.05

Test statistic formula is given as below:

Z = (p̂ - p)/sqrt(pq/n)

We are given x = 36, n = 50

p̂ = x/n = 36/50 = 0.72

p = 0.80,

q = 1 – p = 1 – 0.80 = 0.20

Z = (0.72 – 0.80)/sqrt(0.80*0.20/50)

Z = -1.4142

P-value = 0.1573

(by using z-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

At the 0.05 level of significance, there is insufficient evidence to conclude that age distribution for this sample is significantly different from the distribution for the population of licensed drivers.