Question

David’s iPod has about $10,000$songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of $225$seconds and a standard deviation of $60$seconds.

a. Describe the shape of the sampling distribution of $\overline{x}$for SRSs of size $n=5$from the population of songs on David’s iPod. Justify your answer.

b. Describe the shape of the sampling distribution of $\overline{x}$ for SRSs of size $n=100$ from the population of songs on David’s iPod. Justify your answer.

Short answer

a. The sample mean's sampling distribution is substantially skewed to the right.

b. The sampling distribution of the sample mean is approximately Normal.

Step-by-step solution

Part (a) : Step 1 : Given information

Given :

Mean : $225$seconds

Standard deviation : $60$seconds.

Size :$n:5$

Part (a) : Step 2 : Simplification

The distribution of the population is substantially biased to the right.

$n=5$

The central limit theorem states that if the sample size is big ($30$or more), the sample mean $\overline{x}$sampling distribution is approximately normal.

The central limit theorem cannot be applied since the sample size of$5$is less than $30$.

The sample mean's sampling distribution has the same shape as the population distribution in this example, hence the sample mean's sampling distribution is substantially skewed to the right.

Part (b) : Step 1 : Given information

Given :

Mean : $225$seconds

Standard deviation : $60$seconds.

Size :$n=100$

Part (b) : Step 2 : Simplification

The distribution of the population is substantially biased to the right.

($n=100$)

The central limit theorem states that if the sample size is large enough ($30$or more), the sampling distribution of the sample mean will be close to normal.

Because the sample size of $100$is greater than $30$, the central limit theorem can be used, and the sampling distribution of the sample mean is approximately Normal.