4. 1 / 1 points
a random sample is selected from a population with mean $ \mu = 99 $ and standard deviation $ \sigma = 10 $. for which of the sample sizes would it be reasonable to think that the $ \bar{x} $ sampling distribution is approximately normal in shape? (select all that apply.)
$ n = 12 $
$ n = 17 $
$ n = 41 $
$ n = 60 $
$ n = 120 $
$ n = 480 $
resources
watch it
submit answer
5. - / 4 points
(a) random samples of size $ n $ were selected from a non - normal population with the means and variances given here.
$ n = 14, \mu = 16, \sigma^2 = 4 $
what can be said about the sampling distribution of the sample mean?
since the sample size is $ \text{---select---} $ 30, we $ \text{---select---} $ assume that the distribution of $ \bar{x} $ is approximately normal.
(b) random samples of size $ n $ were selected from a normal population with the means and variances given here.
$ n = 12, \mu = 17, \sigma^2 = 9 $
what can be said about the sampling distribution of the sample mean?
because the original population is normal, even though the sample size is $ \text{---select---} $ 30, we $ \text{---select---} $ assume that the distribution of $ \bar{x} $ is approximately normal.
submit answer
6. 6 / 6 points

Question

Accepted Answer

### Question 4
# Explanation:
To determine when the sampling distribution of the sample mean $\bar{x}$ is approximately normal, we use the Central Limit Theorem (CLT). The CLT states that for a population with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample mean $\bar{x}$ becomes approximately normal as the sample size $n$ increases, regardless of the population distribution (for non - normal populations, a common rule of thumb is $n\geq30$; if the population is normal, the sampling distribution of $\bar{x}$ is normal for any $n$). But in general, for most cases, when $n$ is large (usually $n\geq30$), the sampling distribution of $\bar{x}$ is approximately normal.
- For $n = 12$: $12<30$, so it is not reasonable to assume the sampling distribution of $\bar{x}$ is approximately normal.
- For $n = 17$: $17<30$, so it is not reasonable to assume the sampling distribution of $\bar{x}$ is approximately normal.
- For $n = 41$: $41\geq30$, so it is reasonable to assume the sampling distribution of $\bar{x}$ is approximately normal.
- For $n = 60$: $60\geq30$, so it is reasonable to assume the sampling distribution of $\bar{x}$ is approximately normal.
- For $n = 120$: $120\geq30$, so it is reasonable to assume the sampling distribution of $\bar{x}$ is approximately normal.
- For $n = 480$: $480\geq30$, so it is reasonable to assume the sampling distribution of $\bar{x}$ is approximately normal.

# Answer:
$n = 41$, $n = 60$, $n = 120$, $n = 480$ (the options with these $n$ values should be selected)

### Question 5 (a)
# Explanation:
## Step 1: Analyze the sample size
We are given a non - normal population. The sample size $n = 14$. We compare $n = 14$ with $30$. Since $14<30$, the sample size is less than $30$.
## Step 2: Determine the assumption
For a non - normal population, when the sample size is less than $30$, we cannot assume that the distribution of $\bar{x}$ is approximately normal.

# Answer:
First blank: less than; Second blank: cannot

### Question 5 (b)
# Explanation:
## Step 1: Analyze the sample size
We are given a normal population. The sample size $n = 12$. We compare $n = 12$ with $30$. Since $12<30$, the sample size is less than $30$.
## Step 2: Determine the assumption
For a normal population, the sampling distribution of the sample mean $\bar{x}$ is normal for any sample size $n$. So even though the sample size is less than $30$, we can assume that the distribution of $\bar{x}$ is approximately normal.

# Answer:
First blank: less than; Second blank: can

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 4

Step 1: Analyze the sample size

Step 2: Determine the assumption

Step 1: Analyze the sample size

Step 2: Determine the assumption

Answer:

Question 5 (a)