Questions: A survey of Canadian teens aged 12 to 17 years reported that roughly 75% of them used a fee-based website to download music. You decide to interview a random sample of 15 U.S. teenagers, who you are assuming behave similarly to the Canadian teenagers. What is the mean of the count X who used a fee-based website to download music? (Do not round.) Interpret this value as it pertains to teens and purchasing downloaded music.

Solution Steps

To find the mean \( \mu \) of the count \( X \) who used a fee-based website to download music, we use the formula:

\[ \mu = n \cdot p \]

where:

• \( n = 15 \) (the number of trials, or U.S. teenagers surveyed)
• \( p = 0.75 \) (the probability of success, or the proportion of teens using a fee-based website)

Calculating this gives:

\[ \mu = 15 \cdot 0.75 = 11.25 \]

The mean \( \mu = 11.25 \) indicates that, on average, out of a random sample of 15 U.S. teenagers, we expect approximately \( 11.25 \) of them to have used a fee-based website to download music. This suggests that a significant portion of teenagers are likely to engage in purchasing music online.

In addition to the mean, we can also calculate the variance \( \sigma^2 \) and standard deviation \( \sigma \) using the following formulas:

\[ \sigma^2 = n \cdot p \cdot q \] \[ \sigma = \sqrt{n \cdot p \cdot q} \]

where \( q = 1 - p = 0.25 \).

Calculating these gives:

\[ \sigma^2 = 15 \cdot 0.75 \cdot 0.25 = 2.8125 \] \[ \sigma = \sqrt{2.8125} \approx 1.677 \]

Final Answer