STA3032 Exam 2 - LearnSmart ch4

If x is a value sampled from a normal distribution with mean μ and standard deviation σ, the z-score of x is an item sampled from a normal distribution with mean ______ and standard deviation _______

0, 1

Let X be a binomial random variable, and let a and b be constants with a < b. The normal approximation will be used to compute binomial probabilities. Match each probability to the appropriate area under the normal curve.

1. P(a≤X≤b) = area between a - 0.5 and b + 0.5; 2. P(X≥a) = area to the right of a - 0.5; 3. P(X≤b) = area to the left of b + 0.5; 4. P(X=a) = area between a - 0.5 and a +0.5

Assume that heights of American women aged 18 to 24 are normally distributed with mean μ = 166 cm and standard deviation σ = 6.3 cm. Let z be the value for which the area to its right under the standard normal curve is 0.20. What is the 80th percentile of the heights?

166 + z(6.3)

Stanford-Binet IQ scores are normally distributed in the population with mean μ = 100 and standard deviation σ = 15. How can one compute the proportion of people whose IQs are greater than 120?

Compute z = (120 − 100)/15, then find the area under the standard normal curve to the right of z.

Which of the following is a way to estimate the success probability p associated with a Bernoulli trial?

Conduct n independent trials, and count the number of successes X. Estimate p with X/n

The expression np is equal to which of the following?

Mean of a binomial random variable with parameters n and p.

Let X ∼ Bin(n, p). Assuming n is large enough, which normal distribution approximates the distribution of X ?

N(np,np(1−p))

Twelve students in a classroom of 20 students are females. Four students are chosen at random. Let X be the number of female students chosen. Which of the following statements is true?

P (X = 4) = (12/20)(11/19)(10/18)(9/17)

Let X1 = −1, X2 = 0, and X3 = 1 be a random sample from a normal population with mean μ and variance σ2. Select that all apply.

The mean μ can be estimated as X ̄ = 0; The uncertainty in estimating μ with X can be approximated by s/ sqrt(n)

Minimum daily temperatures in degrees Fahrenheit during the winter in a city in New England are normally distributed with a mean temperature of 27oF and a standard deviation of 5oF. Let F be the temperature in degrees Fahrenheit on a certain day, and let C = (5/9)(F − 32) be the temperature in degrees Celsius. Select all that apply.

The variance of the temperatures in degrees Celsius is σ^2 = (5/9)^2(5)^2; The random variable C = (5/9)(F − 32) also has a normal distribution; C = (5/9)(F − 32) is a linear function of the random variable F .

A coin was tossed 50 times, and in 15 of these tosses a head came up. The probability that a head comes up can be estimated as pˆ = 15/50.

True

A coin with probability 0.7 coming up tails is tossed. Let X = 1 if a head comes up, and X = 0 otherwise. Select all that apply.

X ~ Bernoulli(0.3) and the variance of X is (0.3)(0.7)

A fair six-sided die is rolled. Let X = 1 if an even number comes up, and X = 0 otherwise. Select all that apply:

X ∼ Bernoulli(1/2), The mean of X is μX =1/2

The Central Limit Theorem states that for sufficient large n (in general n > 30) the distribution of the mean X ̄n and the sum Sn of a sample of size n are approximately _______ distributed regardless of the distribution of the population from which these samples are drawn.

normally

Let X ∼ Bin(n, p). Under what conditions can we use the Central Limit Theorem to approximate the distribution of X with a normal distribution?

np>10 and n(1−p)>10

A coin that is suspected to be biased is tossed 100 times, and 81 heads come up. Estimate the probability p that a head comes up and find the uncertainty in pˆ.

pˆ=0.81, The uncertainty in pˆ is sqrt (0.81)(0.19)/100.

For data from a normal population, z-scores are from the ______ normal population

standard

The quantity sqrt( np(1 − p) ) is the ____ of a binomial random variable with parameters n and p

standard deviation

In order to standardize a value x from a normal distribution, we must ___ the mean from x and ___ the result by the standard deviation

subtract, divide

If θˆ is an unbiased estimator, then the mean squared error (MSE) of θˆ is equal to the _____ of θˆ.

variance

What z-score corresponds to the 20th percentile of the normal curve?

z = -0.84