STAT HW#6

Assume that adults have IQ scores that are normally distributed with a mean of μ=100 and a standard deviation σ=15. Find the probability that a randomly selected adult has an IQ less than 115.

0.8413

What are numerical variables with outcomes that cannot be listed or counted because they occur over a range​ called?

Continuous When a numerical variable has outcomes that cannot be listed or counted because they occur over a​ range, the variable is said to be continuous. Next question

Twenty percent of adults in a particular community have at least a​ bachelor's degree. Suppose x is a binomial random variable that counts the number of adults with at least a​ bachelor's degree in a random sample of 100 adults from the community. Which of the following probability statements indicates the probability that fewer than 30 adults have at least a​ bachelor's degree?

P(x<​30)

A teacher wants to find out whether the chance of drawing a Jack is 7.7%. In the last 5 minutes of​ class, he has all the students draw cards, replacing the previous card and shuffling between each draw, until the end of class and then report their results to him. Which​ condition(s) for use of the binomial model​ is/are not​ met?

The number of trials is fixed.

What are the values of the mean and the standard deviation for the standard Normal​ model?

The standard Normal model has a mean of 0 and a standard deviation of 1.

Suppose that a person who has recently helped start a company for the first time is randomly selected. The probability that the company will fail within 4 years is 0.3. Suppose we follow 14 start-ups ​(98 ​people) for 4 years and record the number of people whose company failed. Why is the binomial model inappropriate for finding the probability that at least 8 of these 98 people will no longer be part of a start-up within 4 years​? List all the binomial conditions that are not met.

The trials are independent.

What determines the exact shape of a Normal​ distribution?

The values of the mean and the standard deviation The exact shape of the Normal distribution is determined by the mean and the standard deviation. Next question

Brian wants to go skiing​ tomorrow, but not unless there is between 3 and 5 inches of new snow. According to the weather​ report, any amount of new snow between 1 inch and 6 inches is equally likely. The probability density curve for​ tomorrow's new snow depth is shown. Find the probability that the new snow depth will be within Brian's ideal range. Copy the​ graph, shade the appropriate​ area, and calculate its numerical value to find the probability. The total area is 1.

What is the probability that there will be between 3 and 5 inches of new​ snow? 2/5

For each​ question, find the area to the right of the given​ z-score in a standard Normal distribution. Include an appropriately labeled sketch of the​ N(0,1) curve. Complete parts a through e. a. Find the area to the right of z=6.00. Choose the correct graph below. d. If you had the exact probability for these tail​ proportions, which would be the biggest and which would be the​ smallest? e. Which is equal to the area in part​ b: the area below​ (to the left​ of) z=−12.00 or the area above​ (to the right​ of) z=−12.00​?

a. 0 b. 0 c. 0 d. The area to the right of z=6.00 would be the biggest and the area to the right of z=35.00 would be the smallest. e. The area below z=−12.00 is equal to the area in part b.

A certain state has the highest high school graduation rate of all states at 90​%. a. In a random sample of 10 high school students from the​ state, what is the probability that 9 will​ graduate? b. In a random sample of 10 high school students from the​ state, what is the probability than 8 or fewer will​ graduate? c. What is the probability that at least 9 high school students in our sample of 10 will​ graduate? a. What is the probability that 9 out of 10 high school students will​ graduate?

a. 0.387 b. 0.264 c. 0.736

Assume a standard Normal distribution. Draw a​ well-labeled Normal curve for each part. a. Find the​ z-score that gives a left area of 0.7307. b. Find the​ z-score that gives a left area of 0.1690.

a. 0.61 b. -0.96

What is another name for the expected value of a probability​ distribution?

the mean

The quantitative scores on a test are approximately Normally distributed with a mean of 500 and a standard deviation of 100. On the horizontal axis of the​ graph, indicate the test scores that correspond with the provided​ z-scores. Answer the questions using only your knowledge of the Empirical rule and symmetry. Complete parts a through f. Indicate the test scores that correspond with the provided​ z-scores. Choose the correct answer below. a. Roughly what percentage of students earn quantitative test scores more than​ 500?

z score 1= 600 a. 50% b. 68% c. about 0% d. about 0% e. 95% f. 2.5%

The binomial probability model is useful in many situations with variables of what​ kind?

​Discrete-valued numerical variables

Determine whether the distribution is a discrete probability distribution. If​ not, state why.

​No, because the probabilities do not sum to 1.

Cards are drawn with replacement from a standard deck until a king is drawn. Does this constitute a binomial​ experiment? Why or why​ not?

​No, because there is not a fixed number of trials.

When a certain type of thumbtack is​ flipped, the probability of its landing tip up​ (U) is 0.52 and the probability of its landing tip down​ (D) is 0.48. Suppose you flip two such​ thumbtacks, one at a time. Make a list of all the possible arrangements using U for up and D for down. Find the probabilities of each possible outcome.

There are four​ arrangements; UU,​ UD, DU, and DD. The probabilities are ​P(UU)=0.2704 ​P(UD)=0.2496 ​P(DU)=0.2496 and ​P(DD)=0.2304

In a large​ city, 55​% of people pass the​ drivers' road test. Suppose that every​ day, 300 people independently take the test. Complete parts​ (a) through​ (d) below. a. What is the number of people who are expected to​ pass? b. What is the standard deviation for the number expected to​ pass? c. After a great many​ days, according to the Empirical​ Rule, on about​ 95% of these​ days, the number of people passing will be as low as​ _____ and as high as​ _____. (Hint: Find two standard deviations below and two standard deviations above the​ mean.)

a. 165 b. 9 c. After a great many​ days, according to the Empirical​ Rule, on about​ 95% of these​ days, the number of people passing will be as low as 147 and as high as 183 d. Yes, because 122 is more than 3 standard deviations below the mean.

According to a statistical​ journal, the average lengths of a newborn baby is 19.4 inches with a standard deviation of 0.8 inches. The distribution of lengths is approximately Normal. Use technology or a table to answer parts​ (a) through​ (c) below. For each include an appropriately labeled and shaded Normal curve. a. What is the probability that a newborn baby will have a length of 18 inches or​ less? Select the correct graph below. b. What percentage of newborn babies will be longer than 20 inches​? Select the correct graph below. c. Baby clothes are sold in a​ "newborn" size that fits infants who are between 18 and 20.8 inches long. What percentage of newborn babies will not fit into the​ "newborn" size either because they are too long or too​ short? Select the correct graph below.

a. 4% b. 22.7% c. 8%

Wechsler IQs are approximately Normally distributed with a mean of 100 and a standard deviation of 15. Do not use the Normal table or technology. You may want to label the figure with Empirical Rule probabilities to help you think about this question. Complete parts​ (a) through​ (f). a. Roughly what percentage of people have IQs more than​ 100? b. Roughly what percentage of people have IQs between 100 and​ 115? c. Roughly what percentage of people have IQs below​ 55? d. Roughly what percentage of people have IQs between 70 and​ 130? e. Roughly what percentage of people have IQs above​ 130? f. Roughly what percentage of people have IQs above​ 145?

a. 50% b. 34% c. about 0% d. 95% e. 2.5% f. about 0%

In a large​ city, 60​% of people pass the​ drivers' road test. Suppose that every​ day, 100 people independently take the test. Complete parts​ (a) through​ (d) below. a. What is the number of people who are expected to​ pass? b. What is the standard deviation for the number expected to​ pass? c. After a great many​ days, according to the Empirical​ Rule, on about​ 95% of these​ days, the number of people passing will be as low as​ _____ and as high as​ _____. (Hint: Find two standard deviations below and two standard deviations above the​ mean.)

a. 60 b. 5 c. After a great many​ days, according to the Empirical​ Rule, on about​ 95% of these​ days, the number of people passing will be as low as 50 and as high as 70 d. Yes, because 83 is more than 3 standard deviations above the mean.

ssume college women have heights with the following distribution​ (inches): ​N(60​, 2.1​). Complete parts​ (a) through​ (d) below. a. Find the height at the 75th percentile. b. Find the height at the 25th percentile. c. Find the interquartile range for heights. d. Is the interquartile range larger or smaller than the standard​ deviation?

a. 61.4 b. 58.6 c. 2.8 d. larger

According to data for a​ population, 3-year-old boys have a mean height of 39 inches and a standard deviation of 2 inches. Assume the distribution is approximately Normal. Complete parts a and b. a. Find the percentile measure for a height of 42 inches for a​ 3-year-old boy. If this​ 3-year-old boy grows up to be a man with a height at the same​ percentile, what will his height​ be? Use a population mean of 70 inches and a population standard deviation of 3 inches.

a. 93rd percentile b. 74.5 inches

The Empirical Rule applies rough approximations to probabilities for any​ unimodal, symmetric distribution. But for the Normal distribution we can be more​ precise, as the figure shows. Use the figure and the fact that the Normal curve is symmetric to answer the questions. Do not use a Normal table or technology. Complete parts a through e. a. Roughly what percentage of​ z-scores are between −2 and​ 2? b. Roughly what percentage of​ z-scores are between −3 and​ 3? c. Roughly what percentage of​ z-scores are between −1 and​ 1? d. Roughly what percentage of​ z-scores are more than​ 0? e. Roughly what percentage of​ z-scores are between 1 and​ 2?

a. 95% b. almost all c. 68% d. 50% e. 13.5%

When a certain type of thumbtack is​ flipped, the probability of its landing tip up​ (U) is 0.6 and the probability of its landing tip down​ (D) is 0.4. Suppose you flip two such​ thumbtacks, one at a time. The probability distribution for the possible outcomes of these flips is shown below. a. Find the probability of getting 0​ ups, 1​ up, or 2 ups when flipping two thumbtacks. b. Make a probability distribution graph of this.

a. Find the probability of getting 0​ ups, 1​ up, or 2 ups when flipping two thumbtacks. The probability of 0 ups is 0.16 ​(Type an integer or a decimal. Do not​ round.) Part 2 The probability of exactly 1 up is 0.48 ​(Type an integer or a decimal. Do not​ round.) Part 3 The probability of 2 ups is 0.36 ​(Type an integer or a decimal. Do not​ round.)

For each​ situation, identify the sample size​ n, the probability of a success​ p, and the number of successes x. When asked for the​ probability, state the answer in the form​ b(n,p,x). There is no need to give the numerical value of the probability. Assume the conditions for a binomial experiment are satisfied. Complete parts​ (a) and​ (b) below. a. A 2017 poll found that 52​% of college students were very confident that their major will lead to a good job. If 30 college students are chosen at​ random, what's the probability that 14 of them were very confident their major would lead to a good​ job? Let a success be a college student being very confident their major would lead to a good job. b. A 2017 poll found that 52​% of college students were very confident that their major will lead to a good job. If 30 college students are chosen at​ random, what's the probability that 10 of them are NOT confident that their major would lead to a good​ job? Let a success be a college student not being confident their major would lead to a good job.

a. The probability is ​b(n,p,x)=b(30,0.52,14) b. The probability is ​b(n,p,x)=(b30,0.48,10)

Use a table to find the indicated area under the standard Normal curve. Include an appropriately labeled sketch of the Normal curve and shade the appropriate region. a. Find the area to the left of a​ z-score of −2.18. b. Find the area to the right of a​ z-score of −2.18.

a. The total area to the left of z=−2.18 is 0.0146 b. The total area to the right of z=−2.18 is 0.9854

Determine whether each of the following variables would best be modeled as continuous or discrete. a. The time it takes to fly from City A to City B. b. The time it takes for a light bulb to burn out. c. The distance a baseball travels in the air after being hit. d. The weight of a T-bone steak. e. The number of statistics students now reading a book.

a. continuous b. continuous c. continuous d. continuous e. discrete

Determine whether each of the following variables would best be modeled as continuous or discrete. a. The amount of snowfall. b. The number of statistics students now reading a book. c. The weight of a T-bone steak. d. The time required to download a file from the Internet. e. The floor area of a house.

a. continuous b. discrete c. continuous d. continuous e. continuous

According to a poll done in​ 2010, 62​% of adults aged​ 25-29 had only a cell phone​ (no landline). Assume that two randomly selected adults​ (aged 25-29) are asked whether they have only a cell phone. Complete parts a through e below. a. If a person has only a cell​ phone, we will record Y​ (for Yes). If​ not, we will record N. List all possible sequences of Y and N for this experiment. b. For each​ sequence, find by hand the probability that it will​ occur, assuming each outcome is independent. Select the correct choice below and fill in the answer boxes to complete your choice. c. What is the probability that neither of the two randomly selected adults has only a cell​ phone? d. What is the probability that exactly one person out of two have only a cell​ phone? e. What is the probability that two out of two​ (both) have only a cell​ phone?

a. ​YY, YN,​ NY, NN b. P(YY)= 0.3844, P(YN)= 0.2356, P(NY)= 0.2356, P(NN)= 0.1444 c. 0.1444 d. 0.4712 e. 0.3844