Statistics Final Ch. 4-6

Which of the following statistics are unbiased estimators of population​ parameters?

Sample variance used to estimate a population variance. Sample mean used to estimate a population mean. Sample proportion used to estimate a population proportion.

Assume that we want to find the probability that when five consumers are randomly​ selected, exactly two of them are comfortable with delivery by drones. Also assume that 45​% of consumers are comfortable with the drones​ (based on a​ survey). Identify the values of​ n, x,​ p, and q.

n=5 x=2 p=.45 q=.55

The complement of​ "at least​ one" is​ _______.

none

A continuous random variable has a​ _______ distribution if its values are spread evenly over the range of possibilities.

uniform

Assume that a procedure yields a binomial distribution with n=3 trials and a probability of success of pequals=0.90. Use a binomial probability table to find the probability that the number of successes x is exactly 1.

use the table saved to desktop

When using the​ _______ always be careful to avoid​ double-counting outcomes.

addition rule

Which word is associated with multiplication when computing​ probabilities?

and

In a probability​ histogram, there is a correspondence between​ _______.

area and probability

A​ _______ is any event combining two or more simple events.

compound event

P(A or B) indicates​ _______.

the probability that in a single​ trial, event A​ occurs, event B​ occurs, or they both occur.

The notation P(z<a) donates

the probability that the z-score is less than a.

A. Each of the 3 digits can be any number from 0 to 9.​ Thus, for the first​ digit, there are 10 possible outcomes. Since each digit can be any number from 0 to​ 9, each of the 3 digits has 10 possible outcomes. Apply the fundamental counting rule to determine the number of different selections that are possible. B. What is the probability of​ winning?

A. 10*10*10=1000 B. the number of ways to win divided by the total number of different simple events 1/1000=0.001

What does​ P(B|A) represent?

The probability of event B occurring after it is assumed that event A has already occurred

What does P(M|B) ​represent?

The probability of getting a​ male, given that someone with blue eyes has been selected.

What conditions would produce a negative​ z-score?

a​ z-score corresponding to an area located entirely in the left side of the curve

A certain group of women has a 0.86​% rate of​ red/green color blindness. If a woman is randomly​ selected, what is the probability that she does not have​ red/green color​ blindness?

be sure to convert the percent into a probability by dividing by 100. Then subtract it from 1. 0.9914

A​ _______ probability of an event is a probability obtained with knowledge that some other event has already occurred.

conditional

A​ _______ random variable has infinitely many values associated with measurements.

continuous

The standard deviation of the distribution of sample means is​ _______.

desktop

A​ _______ random variable has either a finite or a countable number of values.

discrete

The​ _______ of a discrete random variable represents the mean value of the outcomes.

expected value

Two events A and B are​ _______ if the occurrence of one does not affect the probability of the occurrence of the other.

independent

Which of the following is a biased​ estimator? That​ is, which of the following does not target the population​ parameter?

median

Identify the expression for calculating the mean of a binomial distribution.

np

"At least​ one" is equivalent to​ _______.

one or more

In a certain weather forecast comma the chance of a thunderstormweather forecast, the chance of a thunderstorm is stated as 18​%. Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.

probability= .18

Which of the following is NOT one of the three methods for finding binomial probabilities that is found in the chapter on discrete probability​ distributions?

A. Use a simulation <-- answer B. Use computer software or a calculator C. Use the binomial probability formula D. Use a statistical table for binomial probabilities

Find the critical value of z subscript 0.07 The expression z Subscript α denotes the z score with an area of α to its right. In this​ case, α equals=0.07.

1-0.07=0.93 invNorm() Z score= 1.48

Events that are​ _______ cannot occur at the same time.

disjoint

Find the area of the shaded region to THE LEFT . The graph depicts the standard normal distribution with mean 0 and standard deviation 1. z=0.83

use 2nd VARS normalcdf() answer=.79673

Find the area of the shaded region TO THE RIGHT. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1. z= -1.07

use 2nd VARS normalcdf() get answer then subtract it from 1 to get final answer: = 1 - 0.1423 =0.8577

Assume that the readings on the thermometers are normally distributed with a mean of 0° and standard deviation of 1.00°C. A thermometer is randomly selected and tested. Draw a sketch and find the temperature reading corresponding to Upper P84​, the 84th percentile. This is the temperature reading separating the bottom 84 % from the top 16 %.

use invNorm() area=.84 =0.99 =99 degrees

.If you bet $ 1.24 in a certain​ state's Pick 4​ game, the expected value is −$0.82. Which bet is​ better, a ​$1.24 bet in the Pick 3 game or a $ 1.24 bet in the Pick 4​ game? Explain.

Neither bet is better because both games have the same expected value.

Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Seven cards are selected from a standard​ 52-card deck without replacement. The number of fours selected is recorded.

No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.

Determine whether the given procedure results in a binomial distribution. If it is not​ binomial, identify the requirements that are not satisfied. Randomly selecting 150 students and recording how much they like rap music on a scale from 1 to 10

No, because there are more than 2 possible outcomes.

Weights of golden retriever dogs are normally distributed. Samples of weights of golden retriever​ dogs, each of size n=​15, are randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be treated as being from a normal distribution because the sample size is too​ small? Explain.

No; the original population is normally​ distributed, so the sample means will be normally distributed for any sample size.

A​ _______ variable is a variable that has a single numerical​ value, determined by​ chance, for each outcome of a procedure.

random

​If, under a given​ assumption, the probability of a particular observed event is extremely​ small, we conclude that the assumption is probably not correct. This represents the​ _______.

rare event rule

P(A) + P(A overbar) right parenthesisPA ​= 1 is one way to express the​ _______.

rule of complementary events

There is a numerical​ (not categorical) random variable​ x, and its number values are associated with corresponding probabilities. ​The sum of all probabilities must be​ 1, but sums such as 0.999 and 1.001 are acceptable because they result from rounding​ errors. Each probability value must be between 0 and 1​ inclusive.

rules for a probability distribution

For the binomial​ distribution, which formula finds the standard​ deviation?

square root of npq

Find the indicated z score WITH AREA TO THE RIGHT. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. area=.1949

subtract area to the right from one before using invNorm() 1 - .1949= .8051 invNorm= .85998

Where would a value separating the top​ 15% from the other values on the graph of a normal distribution be​ found?

the right side of the horizontal scale of the graph

Find the indicated z score TO THE LEFT. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. area= .9761

use 2nd VARS invNorm() =1.979

Which of the following is NOT a descriptor of a normal distribution of a random​ variable?

A. The graph is centered around 0. <--answer B. The graph of the distribution is symmetric. C. The graph is centered around the mean. D. The graph of the distribution is​ bell-shaped.

Which of the following is not true?

A. A​ z-score is an area under the normal curve. <-- Answer B. If values are converted to standard​ z-scores, then procedures for working with all normal distributions are the same as those for the standard normal distribution. C. The area in any normal distribution bounded by some score x is the same as the area bounded by the equivalent​ z-score in the standard normal distribution. D. A​ z-score is a conversion that standardizes any value from a normal distribution to a standard normal distribution.

Which of the following is not a requirement of the binomial probability​ distribution?

A. Each trial must have all outcomes classified into two categories. B. The probability of a success remains the same in all trials. C. The procedure has a fixed number of trials. D. The trials must be dependent. <-- answer

If A denotes some​ event, what does A (overbar on top) ​denote? If ​P(A)equals=0.994​, what is the value of ​P(A (overbar)​)?

Event A (overbar) denotes the complement of event​ A, meaning that A (overbar) consists of all outcomes in which event A does not occur. If ​P(A)equals=0.994​, what is the value of ​P(A overbar)? P(A overbar)= .006

Which of the following is NOT a property of the sampling distribution of the​ variance?

The distribution of sample variances tends to be a normal distribution.

Which of the following is NOT a conclusion of the Central Limit​ Theorem?

The distribution of the sample data will approach a normal distribution as the sample size increases.

Finding probabilities associated with distributions that are standard normal distributions is equivalent to​ _______.

Finding the area of the shaded region representing that probability

Which of the following is NOT a property of the sampling distribution of the sample​ mean?

The distribution of the sample mean tends to be skewed to the right or left.

In the binomial probability​ formula, the variable x represents the​ _______.

Number of successes

Which values cannot be probabilities?

Numbers have to be between 0 and 1. So can't be: negative numbers (ex. -0.41) numbers higher than 1 (ex. 1.5, 5/3)

A researcher collects a simple random sample of​ grade-point averages of statistics​ students, and she calculates the mean of this sample. Under what conditions can that sample mean be treated as a value from a population having a normal​ distribution?

The sample has more than 30​ grade-point averages. If the population of​ grade-point averages has a normal distribution.

Based on a​ survey, assume that 41% of consumers are comfortable having drones deliver their purchases. Suppose we want to find the probability that when six consumers are randomly​ selected, exactly three of them are comfortable with the drones. What is wrong with using the multiplication rule to find the probability of getting threethree consumers comfortable with drones followed by three consumers not​ comfortable?

There are other arrangements consisting of three consumers who are comfortable and three who are not. The probabilities corresponding to those other arrangements should also be included in the result.

Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 90 randomly selected​ individuals, with the number of individuals responding favorably recorded.

Yes, because the experiment satisfies all the criteria for a binomial experiment.

A Gallup poll of 1236 adults showed that​ 12% of the respondents believe that it is bad luck to walk under a ladder. Consider the probability that among 30 randomly selected people from the 1236 who were​ polled, there are at least 2 who have that belief. Given that the subjects surveyed were selected without​ replacement, the events are not independent. Can the probability be found by using the binomial probability​ formula? Why or why​ not?

Yes. Although the selections are not​ independent, they can be treated as being independent by applying the​ 5% guideline.

Selections made with replacement are considered to be​ _______.

independent

Assume that adults have IQ scores that are normally distributed with a mean of μ=105 and a standard deviation σ=15. Find the probability that a randomly selected adult has an IQ less than 135.

.9772

Which of the following does NOT describe the standard normal​ distribution?

A. The total area under the curve must equal 1. B. It is a normal distribution with a mean of 0 and a standard deviation of 1. C. The graph is uniform.<--answer D. The graph is symmetric.

Assume that when adults with smartphones are randomly​ selected, 56​% use them in meetings or classes. If 7 adult smartphone users are randomly​ selected, find the probability that exactly 4 of them use their smartphones in meetings or classes.

Answer= .2932 2nd VARS use binompdf() n=#trials p=probability of success x= the number of successes

As a procedure is repeated again and​ again, the relative frequency of an event tends to approach the actual probability. This is known as​ _______.

As a procedure is repeated again and​ again, the relative frequency of an event tends to approach the actual probability. This is known as the law of large numbers.

Which of the following is not a commonly used​ practice?

If the distribution of the sample means is normally​ distributed, and n greater than >​30, then the population distribution is normally distributed.

The​ _______ tells us that for a population with any​ distribution, the distribution of the sample means approaches a normal distribution as the sample size increases.

The Central Limit Theorem tells us that for a population with any​ distribution, the distribution of the sample means approaches a normal distribution as the sample size increases.

If calculations are​ time-consuming and if a sample size is no more than​ 5% of the size of the​ population, the​ _______ states to treat the selections as being independent​ (even if the selections are technically​ dependent).

The 5% guideline for Cumbersome Calculations

The classical approach to probability requires that the outcomes are​ _______.

The classical approach to probability requires that the outcomes are equally likely.

What requirements are necessary for a normal probability distribution to be a standard normal probability​ distribution?

The mean and standard deviation have the values of μ=0 and.σ=1.

Let event A=subject is telling the truth and event Be=polygraph test indicates that the subject is lying. Use your own words to translate the notation P(B|A) into a verbal statement.

The probability that the polygraph indicates lying given that the subject is actually telling the truth.

The table to the right lists probabilities for the corresponding numbers of girls in three births. What is the random​ variable, what are its possible​ values, and are its values​ numerical? Number of girls (x) P(x) 0 0.125 1 0.375 2 0.375 3 0.125

The random variable is x, which is the number of girls in three births. The possible values of x are 0, 1, 2, and 3. The values of the random value x are numerical.

The​ _______ for a procedure consists of all possible simple events or all outcomes that cannot be broken down any further.

The sample space for a procedure consists of all possible simple events or all outcomes that cannot be broken down any further.

_____________ is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.

The sampling distribution of a statistic

​_____________ is the distribution of sample​ proportions, with all samples having the same sample size n taken from the same population.

The sampling distribution of the proportion

Assume that thermometer readings are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A thermometer is randomly selected and tested. For the case​ below, draw a​ sketch, and find the probability of the reading.​ (The given values are in Celsius​ degrees.) Between 0.50 and 1.75

Use normalcdf() with 0.50 being the lower value and 1.75 being the upper value

Which of the following groups of terms can be used interchangeably when working with normal​ distributions?

areas, probability, and relative frequencies

The conditional probability of B given A can be found by​ _______.

assuming that event A has​ occurred, and then calculating the probability that event B will occur

Determine whether the value is a discrete random​ variable, continuous random​ variable, or not a random variable. a. The number of bald eagles in a country b. The time it takes to fly from City A to City B c. The political party affiliation of adults in the United States d. The time required to download a file from the Internet e. The amount of rain in City B during April f. The number of textbook authors now sitting at a computer

a. It is a discrete random variable. b. It is a continuous random variable c. It is not a random variable. d. It is a continuous random variable e. It is a continuous random variable f. It is a discrete random variable

If you are asked to find the 85th​ percentile, you are being asked to find​ _____.

a data value associated with an area of 0.85 to its left

The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 9 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes.

(9-2.25)/9 = .75

The​ _______ states that​ if, under a given​ assumption, the probability of a particular observed event is exceptionally small​ (such as less than​ 0.05), we conclude that the assumption is probably not correct.

Rare Event Rule for Inferential Statistics

There are​ 15,958,866 adults in a region. If a polling organization randomly selects 1235 adults without​ replacement, are the selections independent or​ dependent? If the selections are​ dependent, can they be treated as independent for the purposes of​ calculations?

The selections are​ dependent, because the selection is done without replacement. Yes, because the sample size is less than​ 5% of the population.

Pulse rates of women are normally distributed with a mean of 77.5 beats per minute and a standard deviation of 11.6 beats per minute. Answer the following questions. What are the values of the mean and standard deviation after converting all pulse rates of women to z scores using z equals StartFraction left parenthesis x minus mu right parenthesis Over sigma EndFractionz= (x−μ) σ​?

μ=0 σ =1 The z scores are numbers without units of measurement.