Week 5 Homework

Suppose X is a normal random variable with mean μ=65 and standard deviation σ=8. ​(a) Compute the​ z-value corresponding to X=55. ​(b) Suppose the area under the standard normal curve to the left of the​ z-value found in part​ (a) is 0.1056. What is the area under the normal curve to the left of X=55​? ​(c) What is the area under the normal curve to the right of X=55​?

(a) z = (x-µ)/σ = (55-65)/8 = -1.25 (b) The area to the left of X=55 is 0.1056 (c) 1 - 0.1056 = 0.8944

According to an​ almanac, 80​% of adult smokers started smoking before turning 18 years old. ​(a) Compute the mean and standard deviation of the random variable​ X, the number of smokers who started smoking before 18 based on a random sample of 300 adults. ​(b) Interpret the mean. (c) Would it be unusual to observe 340 smokers who started smoking before turning 18 years old in a random sample of 400 adult​ smokers? Why?

(a) µx = (300 * 0.80) = 240 σx = √(300 * 0.80)*(1-0.80) = 6.928 (b) It is expected that in a random sample of 300 adult​ smokers, 240 will have started smoking before turning 18. (c) Yes, because 340 is greater than µ + 2σ (More than two standard deviations from the mean)

The area under a normal curve corresponding to a certain characteristic of the normal random variable may be interpreted in any of the following ways:

1. The area corresponds to the probability that a randomly selected individual from the population has the characteristic. 2. The area corresponds to the proportion of the population with the characteristic.

What is the criteria for a binomial experiment?

1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. The outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are two mutually exclusive (disjoint) outcomes: success or failure. 4. The probability of success is fixed for each trial of the experiment.

Find the value of z(0.44)

invNorm ([1-0.44 = 0.56], 0, 1) = 0.15

What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success​ p?

E(x) = np

A binomial experiment is performed a fixed number of times. What is each repetition of the experiment​ called?

Each repetition of the experiment is called a trial.

A continuous random variable has _______ values.

Infinitely many

What value is associated with the inflection points of a normal curve?

One standard deviation above or below the mean.

How do you calculate the probability that between 8 and 10 flights, inclusive, are on time?

P(8≤x≤10) = P(x≤10) - P(x≤[8-1])

What are inflection points?

The points where the curvature of the graph changes.

Determine if the following probability experiment represents a binomial experiment. If​ not, explain why. If the probability experiment is a binomial​ experiment, state the number of​ trials, n. A random sample of 25 professional athletes is​ obtained, and the individuals selected are asked to state their hair length.

This is not a binomial experiment because there are more than two mutually exclusive outcomes for each trial.

Determine whether the distribution is a discrete probability distribution.

Yes, because the sum of the probabilities is equal to 1 and each probability is between 0 and​ 1, inclusive.

Suppose a life insurance company sells a $270,000 ​one-year term life insurance policy to a 24​-year-old female for $220. The probability that the female survives the year is 0.999468. a) Compute and interpret the expected value of this policy to the insurance company. b) Which of the following interpretation of the expected value is​ correct?

a) Probability that she doesn't survive: 1-0.999468 = 0.000532 If she survives: Profit of $220. If she dies: Loss of 220-270000 = -$269,780. Expected value: (Profit * Probability she survives) - ([Potential payout - Premium paid] * Probability she dies) (220*0.999468)-(269780*0.000532) = $76.36 b) The insurance company expects to make an average profit of $76.36 on every 24-year-old female it insures for 1 year.

What do n, p, and 1-p represent in a binomial probability distribution?

n = number of independent trials p = probability of success 1-p = probability of failure

What are the two requirements for a discrete probability​ distribution?

∑P(x)=1 0≤P(x)≤1

Find the​ z-scores that separate the middle 20​% of the distribution from the area in the tails of the standard normal distribution.

100 - 20 = 80 80/2 = 40% in each tail invNorm (0.40, 0, 1) = -0.25 invNorm ([0.40 + 0.20 = 0.60], 0, 1) = 0.25

A study was conducted that resulted in the following relative frequency histogram. Determine whether or not the histogram indicates that a normal distribution could be used as a model for the variable.

The histogram is not bell-shaped, so a normal distribution could not be used as a model for the variable.

The mean incubation time of fertilized eggs is 22 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day. Determine the incubation times that make up the middle 95​%.

The middle 95% is within two standard deviations above or below the mean. 22-1-1 = 20 and 22+1+1 = 24.

What does the area under the graph of a probability density function over an interval represent?

The probability of observing a value of the random variable in that interval.

Determine if the following probability experiment represents a binomial experiment. If​ not, explain why. If the probability experiment is a binomial​ experiment, state the number of​ trials, n. Four cards are selected from a standard​ 52-card deck without replacement. The number of nines selected is recorded.

This is not a binomial experiment because the trials of the experiment are not independent since the probability of success differs from trial to trial.

What is a binomial probability distribution?

A discrete probability distribution that describes probabilities for experiments in which there are two mutually exclusive (disjoint) qualitative outcomes. The two outcomes are generally referred to as success and failure. Experiments in which only two outcomes are possible are referred to as binomial experiments, provided that certain criteria are met.

What happens to a graph as the standard deviation increases or decreases?

It widens and flattens out or narrows and the peak rises.

What value is associated with the peak of a normal curve?

Mean

What are the formulas for the mean (expected value) and standard deviation of a binomial random variable?

Mean: µx = np Standard deviation: σx = square root of np(1-p)

Why can the Empirical Rule be used to identify results in a binomial​ experiment?

The Empirical Rule can be used to identify results in binomial experiments when (1−p)≥10.

In a certain​ state, 45​% of adults indicated that sausage is their favorite pizza. Suppose a simple random sample of adults in the state of size 23 is obtained and the number of adults who indicated that sausage is their favorite pizza was 13. What are values of the parameters​ n, p, and x in the binomial probability​ experiment?

n = 23 p = 0.45 x = 13

Suppose that the probability that a passenger will miss a flight is 0.0945. Airlines do not like flights with empty​ seats, but it is also not desirable to have overbooked flights because passengers must be​ "bumped" from the flight. Suppose that an airplane has a seating capacity of 52 passengers. ​(a) If 54 tickets are​ sold, what is the probability that 53 or 54 passengers show up for the flight resulting in an overbooked​ flight? ​(b) Suppose that 58 tickets are sold. What is the probability that a passenger will have to be​ "bumped"? ​(c) For a plane with seating capacity of 60 ​passengers, how many tickets may be sold to keep the probability of a passenger being​ "bumped" below 5​%?

(a) binompdf (54, [1-0.0945=0.9055], 53) + binompdf (54, 0.9055, 54) = 0.0312 (b) 1 - binomcdf (58, 0.9055, 52) = 0.5281 (c) binompdf (61, 0.9055, 61) = 0.0023 binompdf (62, 0.9055, 62) + binompdf (62, 0.9055, 61) = 0.0159 → still under 5% binompdf (63, 0.9055, 63) + binompdf (63, 0.9055, 62) + binompdf (63, 0.9055, 61) = 0.5547 → over 5%, so the maximum number is 62.

The number of chocolate chips in a bag of chocolate chip cookies is approximately normally distributed with a mean of 1261 chips and a standard deviation of 117 chips. ​(a) Determine the 26th percentile for the number of chocolate chips in a bag. ​(b) Determine the number of chocolate chips in a bag that make up the middle 96​% of bags. ​(c) What is the interquartile range of the number of chocolate chips in a bag of chocolate chip​ cookies?

(a) invNorm (0.26, 1261, 117) = 1186 (b) invNorm (0.02, 1261, 117) = 1021 invNorm (0.98, 1261, 117) = 1501 (c) invNorm (0.25, 1261, 117) = 1182 invNorm (0.75, 1261, 117) = 1340 1340 - 1182 = 158

State the seven properties of a normal curve.

1. The normal curve is symmetric about its mean, µ. 2. Because mean=median=mode, the normal curve has a single peak and the highest point occurs at x = µ. 3. The normal curve has inflection points at µ-σ and µ+σ. 4. The area under the normal curve is 1. 5. The area under the normal curve to the right of µ equals the area under the normal curve to the left of µ, which equals 1/2. 6. As x increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis. As x decreases without bond (becomes more and more negative), the graph approaches, but never reaches, the horizontal axis. 7. The Empirical Rule: Approximately 68% of the area under the normal curve is between x = µ - σ and x = µ + σ, approximately 95% of the area is between x = µ - 2σ and x = µ + 2σ, and approximately 99.7% of the area is between x = µ - 3σ and x = µ + 3σ.

There are two college entrance exams that are often taken by​ students, Exam A and Exam B. The composite score on Exam A is approximately normally distributed with mean 20.6 and standard deviation 5.2. The composite score on Exam B is approximately normally distributed with mean 1016 and standard deviation 209. Suppose you scored 23 on Exam A and 1243 on Exam B. Which exam did you score better​ on? Justify your reasoning using the normal model.

Exam A: normalcdf (-9999999, 27, 20.6, 5.2) = 0.91 Exam B: normalcdf (-9999999, 1243, 1016, 209) = 0.86 The score on Exam A is​ better, because the percentile for the Exam A score is higher.

One graph in the figure represents a normal distribution with mean μ=8 and standard deviation σ=2. The other graph represents a normal distribution with mean μ=14 and standard deviation σ=2. Determine which graph is which and explain how you know.

Graph A has a mean of μ=8 and graph B has a mean of μ=14 because a larger mean shifts the graph to the right.

What does it mean to say that a continuous random variable is normally distributed?

It is bell-shaped.

What happens to a graph as the mean increases or decreases?

It shifts right or shifts left.

Determine the area under the standard normal curve that lies to the left of: (a) Z = 0.81 ​(b) Z = 1.39​ ​(c) Z = 0.01​ (d) Z = 0.95

Standard normal curve always has µ = 0 and σ = 1 For area to the left: (a) normalcdf (-9999999, 0.81, 0, 1) = 0.7910 (b) normalcdf (-9999999, 1.39, 0, 1) = 0.9177 (c) normalcdf (-9999999, 0.01, 0, 1) = 0.5040 (d) normalcdf (-9999999, 0.95, 0, 1) = 0.8289

When can the Empirical Rule be used to identify unusual results in a binomial​ experiment?

When the binomial distribution is approximately bell​ shaped, about​ 95% of the outcomes will be in the interval from μ−2σ to μ+2σ.

Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​a) The number of light bulbs that burn out in the next week in a room with 16 bulbs. ​b) The time it takes for a light bulb to burn out.

a) The random variable is discrete. The possible values are x=​0, 1, ​2, ...16. b) The random variable is continuous. The possible values are t>0.

In the probability distribution to the​ right, the random variable X represents the number of marriages an individual aged 15 years or older has been involved in. a) Compute and interpret the mean of the random variable X. b) Which of the following interpretations of the mean is​ correct?

a) µx=0.928 marriages b) If many individuals aged 15 year or older were​ surveyed, the sample mean number of marriages should be close to the mean of the random variable.