Q: A uniform random variable $$X on$$ $$[a, b]$$ has the following probability density function:

$$f(x) = \dfrac{1}{b-a}$$ \,\text{for}\, a \leq x \leq b$

Question 1

Which of the following statements about a continuous random variable is TRUE?

Accepted Answer

Its possible values form an interval and can take infinitely many values.

Question 2

The total area under a density curve for a continuous random variable is:

Accepted Answer

$$1$$

Question 3

Suppose $$X is a $$continuous random variable with density function $$f(x). $$What does $$P(a \u003c X \u003c b)$$ represent?

Accepted Answer

The area under the density curve between $$x = a$$ and $$x = b$$

Question 4

A uniform random variable $$X on$$ $$[a, b]$$ has the following probability density function:

Accepted Answer

$$f(x) = \dfrac{1}{b-a}$$ \,	ext{for}\, a \leq x \leq b$

Question 5

The mean and standard deviation of a uniform random variable $$X on$$ $$[a, b]$$ are given by:

Accepted Answer

$$\mu = \dfrac{a+b}{2}$$ and $$\sigma = \sqrt{\dfrac{(b-a)^2$}{12}}$$

Question 6

If $$X is a $$normal random variable with mean $$\mu$$ and standard deviation $$\sigma$$, which of the following is TRUE?

Accepted Answer

The mean, median, and mode of $$X$$ are all equal to $$\mu.$$

Question 7

The standard normal random variable $$Z$$ has which parameters?

Accepted Answer

$$\mu = 0$$, $$\sigma = 1$$

Question 8

A test score $$X is $$normally distributed with $$\mu = 80$$ and $$\sigma = 10. $$What is the $$z-$$score for a student who scored $$95$$?

Accepted Answer

$$z = \dfrac{95 - 80}{10} = 1.5$$

Question 9

According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

$$95.44\%$$

Question 10

A population has mean $$\mu = 100$$ and standard deviation $$\sigma = 20. $$What is the standard deviation of the sample mean $$\overline{X}$$ for samples of size $$n = 25$$?

Accepted Answer

$$\sigma_{\overline{X}} = \dfrac{20}{\sqrt{25}} = 4$$

Question 11

Which statement best describes the Central Limit Theorem?

Accepted Answer

For large sample sizes, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

Question 12

A population proportion $$p is $$estimated by the sample proportion $$\hat{p}. $$Which formula is used to calculate $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{x}{n}$$, where $$x is $$the number with the attribute and $$n is $$the sample size.

Question 13

The standard deviation of the sample proportion $$\hat{p}$$ for large $$n is$$:

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$$

Question 14

If the IQ scores in a population are normally distributed with $$\mu = 100$$ and $$\sigma = 16$$, what is the IQ score at the 90th percentile? (Use $$z_{0.90} \approx 1.28)$$

Accepted Answer

$$x = 100 + 16 	imes 1.28 = 120.48$$

Question 15

A sample of size $$n is $$taken from a population with mean $$\mu$$ and standard deviation $$\sigma. $$What is the mean of the sampling distribution of the sample mean $$\overline{X}$$?

Accepted Answer

$$\mu_{\overline{X}} = \mu$$

Question 16

Which of the following is an unbiased estimator for the population mean $$\mu$$?

Accepted Answer

The sample mean $$\overline{X}$$

Question 17

A pharmaceutical company wants to estimate the proportion of aspirin tablets that contain at least 700 mg of pain reliever. In a random sample of 100 tablets, 92 contain at least 700 mg. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{92}{100} = 0.92$$

Question 18

If $$X is a $$continuous random variable, what is $$P(X = c)$$ for any real number $$c$$?

Accepted Answer

$$0$$

Normal Probability Distributions and Sampling Distributions: Study Notes

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