The binomial distribution

Model repeated independent trials using the binomial distribution; calculate and interpret probabilities; find expected value and variance; use the normal approximation for large samples and understand the sampling distribution of the mean.

Statistical Analysis · ME-12-06

Sub-topics

The binomial distribution

Recognise Bernoulli trials; calculate binomial probabilities using

P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}

; find the mean

np

and variance

np(1-p)

of a binomial distribution.

Sampling distribution of the mean

Understand the sample mean and sample proportion as random variables; apply the normal approximation to the binomial distribution; use the sampling distribution of

\bar{X}

with mean

\mu

and standard error

\sigma/\sqrt{n}

Open Math

The binomial distribution

Statistical Analysis · ME-12-06

Name:

Date:

The binomial distribution

Q1Straightforward

A fair coin is tossed 5 times. Let

X

be the number of heads. Find

P(X = 3)

Q2Straightforward

X

follows a binomial distribution with parameters

n = 8

and

p = 0.5

. Find

E(X)

Q3Straightforward

X \sim B(10, 0.3)

. Find

\text{Var}(X)

Q4Straightforward

A factory produces items, 20% of which are defective. A random sample of 4 items is selected. Find the probability that exactly 2 items are defective.

Q5Moderate

A biased die is rolled 6 times. The probability of rolling a six on any single roll is

\dfrac{1}{3}

. Find the probability of rolling no sixes. Give your answer to 4 decimal places.

Q6Moderate

X \sim B(12, 0.25)

. Find the standard deviation of

X

Q7Moderate

A basketball player makes 70% of their free throws. In a game they attempt 5 free throws. Find the probability of making exactly 4. Give your answer to 4 decimal places.

Q8Moderate

X \sim B(n, p)

has mean

6

and variance

2.4

. Find

n

Q9Moderate

In a quality control inspection, 15% of products are found to be defective. A batch of 10 products is tested. Find the probability that at most 1 product is defective. Give your answer to 4 decimal places.

Q10Moderate

A fair coin is tossed 8 times. Find the probability of getting at least 1 head. Give your answer to 4 decimal places.

Q11Challenging

A fair coin is tossed 8 times. Find the probability of getting at least 6 heads. Express your answer as a fraction in simplest form.

Q12Challenging

The probability of success on a single trial is

0.1

. Find the minimum number of independent trials needed so that the probability of at least one success exceeds

0.5

Q13Challenging

A production process has a 90% pass rate. A sample of 10 items is tested. Find the probability that fewer than 9 items pass. Give your answer to 4 decimal places.

Sampling distribution of the mean

Q14Straightforward

X \sim B(25, 0.6)

. When the binomial distribution is approximated by a normal distribution, what is the mean of the approximating normal distribution?

Q15Straightforward

X \sim B(25, 0.5)

. When the binomial distribution is approximated by a normal distribution, what is the standard deviation of the approximating normal distribution?

Q16Straightforward

A random sample of size

n = 64

is taken from a population with standard deviation

\sigma = 8

. Find the standard error of the sample mean

\bar{X}

Q17Straightforward

X \sim B(100, 0.5)

is approximated by

N(50, 25)

(mean 50, variance 25). Find the

z

-score corresponding to

X = 60

Q18Moderate

A random sample of

n = 100

observations is drawn from a population with mean

\mu = 50

and standard deviation

\sigma = 20

. Find the standard error of the sample mean

\bar{X}

Q19Moderate

X \sim B(400, 0.5)

is approximated by

N(200, 100)

(mean 200, standard deviation 10). Using the empirical rule (68–95–99.7 rule), find

P(190 < X < 210)

Q20Moderate

A random sample of size

n = 36

is taken from a population with mean

\mu = 80

and standard deviation

\sigma = 12

. The sample mean

\bar{X}

is approximately

N(80, 4)

(mean 80, standard deviation 2). Using the empirical rule, find

P(\bar{X} > 82)

Q21Moderate

The true proportion of voters who support a candidate is

p = 0.3

. A random sample of

n = 100

voters is surveyed. The standard error of the sample proportion

\hat{p}

\text{SE}(\hat{p}) = \sqrt{p(1-p)/n}

. Find

\text{SE}(\hat{p})

to 4 decimal places.

Q22Moderate

X \sim B(196, 0.5)

is approximated by

N(98, 49)

(mean 98, standard deviation 7). Using the empirical rule, find

P(X < 105)

Q23Challenging

Samples of size

n = 25

are repeatedly taken from a population with mean

\mu = 70

and standard deviation

\sigma = 10

. The sample mean

\bar{X}

is approximately normal with standard error

\text{SE} = 2

. Using the empirical rule, find

P(|\bar{X} - 70| > 4)

Q24Challenging

For

X \sim B(n, 0.4)

, the normal approximation is valid when both

np \geq 5

and

n(1-p) \geq 5

. Find the minimum value of

n

for which both conditions are satisfied.

Q25Challenging

In a large population, the true proportion who prefer a new product is

p = 0.5

. A random sample of

n = 100

people is surveyed, and the sample proportion

\hat{p}

has standard error

\text{SE} = 0.05

. Using the empirical rule, find

P(0.40 \leq \hat{p} \leq 0.60)

Worked solutions and answers at openmath.au/year-12/extension-1/the-binomial-distribution