Q: Suppose the probability of a rare disease in a population is 0.001. In a random sample of 1000 people, what is the expected number of people with the disease?

$$ E(X) = 1000 \times 0.001 = 1 $$

Q: A researcher wants to estimate the mean cholesterol level in a population. What sample size is needed to estimate the mean within 5 units with 95% confidence, assuming $$ \sigma = 20 $$? (Use $$ z_{\alpha/2} = 1.96 .)$$

$$ n = \left( \frac{1.96 \times 20}{5} \$$right)^2$$ = 61.47 $$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable whose possible values are countable and distinct, such as the number of heads in 10 coin tosses.

Question 2

A fair die is rolled 4 times. What is the probability of getting exactly two sixes? Use the binomial probability formula.

Accepted Answer

$$ P(X=2) = \binom{4}{2} \left(\frac{1}{6}\$$right)^2$$ \left(\frac{5}{6}\$$right)^2$$ $$

Question 3

Which of the following is NOT a property of a normal distribution?

Accepted Answer

The curve is skewed to the right.

Question 4

Suppose the probability of a rare disease in a population is 0.001. In a random sample of 1000 people, what is the expected number of people with the disease?

Accepted Answer

$$ E(X) = 1000 	imes 0.001 = 1 $$

Question 5

A continuous uniform distribution is defined on the interval $$ [a, b] . $$What is the probability that a randomly selected value falls between $$ c $$ and $$ d $$, where $$ a \leq c \u003c d \leq b $$?

Accepted Answer

$$ P(c \u003c X \u003c d) = \frac{d-c}{b-a} $$

Question 6

If a normal distribution has mean $$ \mu = 50 $$ and standard deviation $$ \sigma = 5 $$, what is the $$ z -$$score for $$ x = 60 $$?

Accepted Answer

$$ z = \frac{60-50}{5} = 2 $$

Question 7

Which of the following statements about the Central Limit Theorem (CLT) is correct?

Accepted Answer

The CLT states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 8

A 95% confidence interval for a population mean is calculated as $$ (12.3, 15.7) . $$Which of the following interpretations is correct?

Accepted Answer

We are 95% confident that the population mean lies between 12.3 and 15.7.

Question 9

A medical test for a disease has a 98% accuracy rate. If 5 people are tested, what is the probability that all 5 tests are accurate? Assume independence.

Accepted Answer

$$ P = 0.98^5$ $$

Question 10

Which of the following is a requirement for a probability distribution to be considered binomial?

Accepted Answer

All of the above.

Question 11

A sample of 36 students has a mean test score of 80 and a standard deviation of 12. What is the standard error of the mean?

Accepted Answer

$$ SE = \frac{12}{\sqrt{36}} = 2 $$

Question 12

If the area to the left of a $$ z -$$score is 0.975, what is the value of $$ z $$? (Use the standard normal distribution.)

Accepted Answer

$$ z = 1.96 $$

Question 13

A researcher wants to estimate the mean cholesterol level in a population. What sample size is needed to estimate the mean within 5 units with 95% confidence, assuming $$ \sigma = 20 $$? (Use $$ z_{\alpha/2} = 1.96 .)$$

Accepted Answer

$$ n = \left( \frac{1.96 	imes 20}{5} \$$right)^2$$ = 61.47 $$

Question 14

A normal distribution has $$ \mu = 100 $$ and $$ \sigma = 15 . $$What is the probability that a randomly selected value is between 85 and 115?

Accepted Answer

$$ P(85 \u003c X \u003c 115) = P(-1 \u003c z \u003c 1) = 0.6826 $$

Question 15

A lottery offers a $$1,000,000$$ prize with a probability of $$1/10,000,000$$ per ticket. If a ticket costs $$2$$, what is the expected value of buying one ticket?

Accepted Answer

$$ E(X) = (1,000,000 	imes \frac{1}{10,000,000}) + (-2 	imes \frac{9,999,999}{10,000,000}) = -1.90 $$

Question 16

Which of the following is true about the standard normal distribution?

Accepted Answer

It has a mean of 0 and a standard deviation of 1.

Question 17

A sample of 25 light bulbs has a mean lifetime of 1200 hours and a standard deviation of 100 hours. What is the 90% confidence interval for the mean lifetime? (Use $$ z_{\alpha/2} = 1.645 .)$$

Accepted Answer

$$ 1200 \pm 1.645 	imes \frac{100}{\sqrt{25}} = 1200 \pm 32.9 $$

Question 18

A quality control manager wants to know if a machine is producing bolts with an average length different from 5 cm. Which type of hypothesis test should be used?

Accepted Answer

A two-tailed test for the population mean.

Question 19

If the probability of an event is 0.03, is the event considered statistically significant at the 0.05 significance level?

Accepted Answer

Yes, because $$ 0.03 \u003c 0.05 .$$

Step-by-Step Guidance for STA 2023 Exam 2 Topics (Chapters 5–7)

Guida di studio - Domande di pratica

Study guide - Flashcard