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%)

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{(p_1(1 - p_1))/(n_1) + (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{(0.001(1 - 0.001))/(10,000) + (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%)