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

Answer:

The standard error of the estimated difference is of 0.0313.

Step-by-step 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 $s = \frac{\sigma}{\sqrt{n}}$.

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 $s = \sqrt{\frac{p(1-p)}{n}}$

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:

$p_A = \frac{33}{197} = 0.1675$

$s_A = \sqrt{\frac{0.1675*0.8325}{197}} = 0.0266$

Brand B:

25 out of 290, so:

$p_B = \frac{25}{290} = 0.0862$

$s_B = \sqrt{\frac{0.0862*0.9138}{290}} = 0.0165$

What would be the standard error of the estimated difference?

$s = \sqrt{s_A^2+s_B^2} = \sqrt{0.0266^2+0.0165^2} = 0.0313$

The standard error of the estimated difference is of 0.0313.