Question

Electric vehicles make up a very small proportion of the overall car market, but by how much has that proportion increased? Independent surveys of randomly selected car dealerships were completed, with 0.1% of the 10,000 cars sold in the 2012 sample being electric vehicles, and 0.5% percent of the 20,000 cars in the 2016 sample being electric vehicles. Which of the following represents a 95% confidence interval for the change in the proportion of electric cars sold from 2012 to 2016?

Answer 1

Given Information:

The proportion of electric cars sold in 2012 = p₁ = 0.1%

The proportion of electric cars sold in 2016 = p₂ = 0.5%

Sample size of electric cars in 2012 = n₁ = 10,000

Sample size of electric cars in 2016 = n₂ = 20,000

Confidence level = 95%

Required Information:

Change in the proportion of electric cars sold from 2012 to 2016 = ?

Answer:

Change in the proportion of electric cars sold from 2012 to 2016 = (0.284%, 0.516%)

Step-by-step explanation:

The change in the mean is given by

$\mu = p_2 - p_1\\\\\mu = 0.005 -0.001\\\\\mu = 0.004$

The change in standard error is given by

$SE = z\cdot\sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2} }$

Where z is the value of z-score corresponding to 95% confidence level.

$z_{\alpha/2} = 1 - 0.95 = 0.05/2 = 0.025\\\\z_{0.025} = 1.96$

$SE = 1.96\cdot\sqrt{\frac{0.001(1 - 0.001)}{10,000} + \frac{0.005(1 - 0.005)}{20,000} } \\\\SE = 1.96\cdot 0.00059\\\\SE = 0.00116$

The change in the proportion of electric cars is

$\mu \pm SE \\\\0.004\pm 0.00116 \\\\(0.004-0.00116) \: and \: (0.004-0.00116)\\\\(0.00284, \: 0.00516)\\\\(0.284\%, \: 0.516\%)$

Therefore, we are 95% confident that the change in the proportion of electric cars sold from 2012 to 2016 is between (0.284%, 0.516%)