2.

Select the correct answer.

The national apple growers organization recently released its first crop of a new apple variety. It gathered data on the weight of the new apples.

It found a population mean of 4.85 ounces and a standard deviation of 0.92. Each sample size was 500 apples. By the central limit theorem,

which interval do 99.7% of the sample means fall within?

OA.

4.81 and 4.89

OB.

4.73 and 4.97

Ос. .

4.84 and 4.86

OD

4.77 and 4.93

Answer:

B. 4.73 and 4.97

Step-by-step explanation:

To solve this question, we need to understand the Empirical Rule and the Central Limit Theorem.

Empirical Rule:

The Empirical Rule states that, for a normally distributed random variable:

Approximately 68% of the measures are within 1 standard deviation of the mean.

Approximately 95% of the measures are within 2 standard deviations of the mean.

Approximately 99.7% of the measures are within 3 standard deviations of the mean.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Standard deviation of 0.92, sample of 500:

This means that [tex]\sigma = 0.92, n = 500, s = \frac{0.92}{\sqrt{500}} = 0.04[/tex]

By the central limit theorem, which interval do 99.7% of the sample means fall within?

Within 3 standard deviations of the mean. So

4.85 - 3*0.04 = 4.85 - 0.12 = 4.73

4.85 + 3*0.04 = 4.85 + 0.12 = 4.97

So, option B.

Answer:

4.73 and 4.97

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