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.

Given :

Mean : $225$seconds

Standard deviation : $60$seconds.

Size :$n:5$

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.

Given :

Mean : $225$seconds

Standard deviation : $60$seconds.

Size :$n=100$

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.