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david’s phone has about 10,000 songs. the distribution of play time for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. suppose we choose an srs of 10 songs from this population and calculate the mean play time $\bar{x}$ of these songs.
(a) calculate the mean and standard deviation of the sampling distribution of $\bar{x}$.
$\mu_{\bar{x}} = $ seconds (do not round.)
$\sigma_{\bar{x}} = $ seconds (round to 2 decimal places.)
(b) interpret the standard deviation from part (a).
in $\text{srss}$ of size $n = 10$ from david’s ipod, the sample mean play time of songs typically varies from $\text{the true mean}$ by about $18.97$ seconds.

Question

Accepted Answer

### Part (a)
# Explanation:
## Step 1: Mean of Sampling Distribution
The mean of the sampling distribution of the sample mean $\bar{x}$ (denoted as $\mu_{\bar{x}}$) is equal to the population mean $\mu$. Given the population mean $\mu = 225$ seconds, we have:
$\mu_{\bar{x}}=\mu = 225$

## Step 2: Standard Deviation of Sampling Distribution
The standard deviation of the sampling distribution of the sample mean (also called the standard error) is given by the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Here, $\sigma = 60$ seconds and $n = 10$. Plugging in the values:
$\sigma_{\bar{x}}=\frac{60}{\sqrt{10}}$
Calculating $\sqrt{10}\approx3.1623$, so $\frac{60}{3.1623}\approx18.97$ (rounded to 2 decimal places).

# Answer:
For $\mu_{\bar{x}}$: $\boldsymbol{225}$ seconds
For $\sigma_{\bar{x}}$: $\boldsymbol{18.97}$ seconds

### Part (b)
# Brief Explanations:
The standard deviation of the sampling distribution (standard error) tells us how much the sample mean $\bar{x}$ varies from the true population mean, on average, when taking samples of size $n = 10$. So, in SRSs (simple random samples) of size $n = 10$ from David’s phone, the sample mean play time of songs typically varies from the true mean (which is 225 seconds) by about 18.97 seconds.

# Answer:
In $\boldsymbol{\text{SRSs}}$ of size $n = 10$ from David’s iPod, the sample mean play time of songs typically varies from $\boldsymbol{\text{the true mean}}$ by about $\boldsymbol{18.97}$ seconds.

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Mean of Sampling Distribution

Step 2: Standard Deviation of Sampling Distribution

Answer:

Part (b)