Question

A consumer advocate agency is concerned about reported failures of two brands of MP3 players, which we will label Brand A and Brand B. In a random sample of 197 Brand A players, 33 units failed within 1 year of purchase. Of the 290 Brand B players, 25 units were reported to have failed within the first year following purchase. The agency is interested in the difference between the population proportions, , for the two brands. Using the data from the two brands, what would be the standard error of the estimated difference, Dp = A – B, if it were believed that the two population proportions were, in fact, equal (i.e., )?

Answer 1

The standard error of the estimated difference is of 0.0313.

Explanation:

To solve this question, we need to understand the central limit theorem, and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

Subtraction between normal variables:

When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

Brand A:

33 out of 197, so:

`p_A = (33)/(197) = 0.1675`

`s_A = \sqrt{(0.1675*0.8325)/(197)} = 0.0266`

Brand B:

25 out of 290, so:

`p_B = (25)/(290) = 0.0862`

`s_B = \sqrt{(0.0862*0.9138)/(290)} = 0.0165`

What would be the standard error of the estimated difference?

`s = √(s_A^2+s_B^2) = √(0.0266^2+0.0165^2) = 0.0313`

The standard error of the estimated difference is of 0.0313.