Question

In a random sample of 85 cars driven at high altitudes, 21 of them exceeded the pollution limits. In an independent study of 340 cars driven at low altitudes, 46 of them exceeded the pollution limits. Can we conclude that the proportion of high altitude vehicles exceeding pollution limits is greater than that for low altitude vehicles? Test at the alpha=0.01 level. Please show all work/calculations. a) State and check the conditions that must be met in order to perform this test. If needed, include a sketch of any appropriate graphs. b) State the hypothesis. Clearly label the population parameter(s). c) Perform the test and state the test used, the P-value, and the decision. Round the P-value to 4 decimal places. Please show all work/calculations d) Write a conclusion using a complete sentence.

Answer 1

Solution:

Part a

For the given scenario, both sample sizes are adequate to use normal distribution. Both sample sizes are greater than 30. Also, we know that the sampling distribution of the difference between the two sample proportions follows an approximate normal distribution.

Part b

Here, we have to use z test for difference between two population proportions.

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The proportion of high altitude vehicles exceeding pollution limits is same as that for low altitude vehicles.

Alternative hypothesis: H_a: The proportion of high altitude vehicles exceeding pollution limits is greater than that for low altitude vehicles.

H₀: p₁ = p₂ versus H_a: p₁ > p₂

p₁ = population proportion of cars at high altitudes which exceeds pollution limits.

P₂ = population proportion of cars at low altitudes which exceeds pollution limits.

We are given level of significance = α = 0.01

X1 = 21, n1 = 85, x2 = 46, n2 = 340

Part c

Test statistic for z test for two population proportions is given as below:

Z = (P1 – P2) / sqrt(P*(1 – P)*((1/N1) + (1/N2)))

Where,

First sample proportion = P₁ = X1/n1 = 21/85 = 0.247058824

Second sample proportion = P₂ = X2/n2 = 46/340 = 0.135294118

First sample size = n₁= 85

Second sample size = n₂ = 340

P = (X1+X2)/(N1+N2) = (21 + 46) / (85 + 340) = 0.1576

Z = (P1 – P2) / sqrt(P*(1 – P)*((1/N1) + (1/N2)))

Z = (0.247058824 – 0.135294118) / sqrt(0.1576*(1 – 0.1576)*((1/85) + (1/340)))

Z = 2.5291

P-value = 0.0057

(by using z-table)

P-value < α = 0.01

So, we reject the null hypothesis

Part d

There is sufficient evidence to conclude that the proportion of high altitude vehicles exceeding pollution limits is greater than that for low altitude vehicles.