Question

practice: central limit theorem
use the word bank to fill in the blank for each statement.
population statistic random parameter sample

1. the mean from a random sample is an example of a

__________________________.

1. a simple ______ sample of size 250 is taken from the student population

of lander university.

1. the proportion of seniors at carolina high school is a value between 0 and 1.

we call this value a ______ because it describes a population.

1. the sampling distribution mean will always be the same as the

__________________________ mean.
for each question, show the work necessary to determine the answer even if you
use a calculator.

1. suppose we have an srs of size n = 60 for a population with a known mean of

8.2 and standard deviation of 1.4. what is the mean of the sampling distribution?
a coffee shop has an average customer wait time of 6 minutes with a standard
deviation of 2.5 minutes. what is the probability that the average wait time for a
systematic random sample of 40 customers exceeds 7 minutes?

Explanation:

Step1: Recall the definitions

• A statistic is a value calculated from a sample.
• A random sample is a sample chosen in a way that every member of the population has an equal chance of being selected.
• A parameter is a value that describes a population.
• The sampling distribution of the sample mean has a mean equal to the population mean.

Step2: Answer the fill - in - the - blank questions

1. The mean from a random sample is an example of a statistic because it is calculated from a sample.
2. A simple random sample of size 250 is taken from the student population of Lander University. This is the standard term for a sample where each individual has an equal chance of being selected.
3. The proportion of seniors at Carolina High School is a value between 0 and 1. We call this value a parameter because it describes a population (all seniors at Carolina High School).
4. The sampling distribution mean will always be the same as the population mean. This is a fundamental property of the sampling distribution of the sample mean.

Step3: Solve problem 5

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). Given \(\mu = 8.2\), the mean of the sampling distribution is \(8.2\).

Step4: Solve problem 6

Let \(X\) be the customer wait time. We know \(\mu = 6\), \(\sigma=2.5\), and \(n = 40\). The standard deviation of the sampling distribution \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{2.5}{\sqrt{40}}\approx0.395\).
We want to find \(P(\bar{X}>7)\). First, calculate the z - score: \(z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}=\frac{7 - 6}{0.395}\approx2.53\).
Using the standard normal distribution \(P(Z>2.53)=1 - P(Z\leq2.53)\). From the standard normal table, \(P(Z\leq2.53)=0.9943\). So \(P(Z>2.53)=1 - 0.9943 = 0.0057\).

Answer:

1. statistic
2. random
3. parameter
4. population
5. \(8.2\)
6. \(0.0057\)