Stat Chapter 15

central limit theorem

says that when n is large, the sampling distribution of the sample mean is approximately Normal: sample mean is approx N(population mean, population standard deviation/square root n)

This tells us the number of standard deviations our value is away from the mean.

z = (our value - mean)/appropriate SD

sample

If we want to estimate the population mean, we use the __________ mean. This mean is a random variable.

usually n > 40

How large is "large enough"?

population

A measurement on every individual of interest is the variable of a(n) __________ distribution.

The measurement of each individual in a population.

What is the variable of a population distribution?

True. The standard deviation of the sampling distribution of x¯ equals σ/square root n. Thus, σ, which measures the variability of individuals, is greater than σ/square root n, which measures the variability of sample means.

True or false: Individual observations are more variable than sample means from large random samples of the same size from the same population.

parameter. It is a parameter because it is a number that describes the entire population, the population being all homes listed in this city for July 2011.

When examining the real estate records of a city in Utah in July 2011, we find the average listing price of all homes in that city was about $230,000. The number 230,000 is a __________.

parameter. A parameter is a number that describes the entire population. It is not a number describing the individuals.

A number describing an entire population is called a _____________.

law of large numbers

Draw observations at random from any population with finite mean μ. As the number of observations drawn increases, the sample mean of the observed values gets closer and closer to the mean μ of the population.

Each random variable has a probability model which tells us the values the random variable can take and the probability that it takes on these values.

Each random variable has a probability model which tells us the values the random variable can take and the probability that it takes on these values. Copy.

simulation

Typically we take just one sample and then generalize to the population as a whole. Before we do that, we need to understand how sample mean behaves. This can be done through ______________.

True. This is a statement of the law of large numbers.

True or false: As we increase the sample size, the sample mean, x¯, gets closer and closer to the population mean, μ.

False. As the sample size increases, the sample mean gets closer and closer to μ with less variability, not more.

True or false: As we increase the sample size, the sample mean, x¯, gets more and more variable.

gets closer to population mean

According to the law of large numbers, as we increase the sample size, the sample mean....

statistic

In 2010 the Consumer Population Survey found their sample of women had a median weekly income of $669. The number $669 is a _____________.

large enough sample

Is our sample size n at least 40?

A. Distribution of data in a sample The distribution of data in a sample is the values of scores for students in a sample.

Which of the following is the statistical name for the comprehensive scores of a sample of 500 students taking the ACT exam (a national college admissions exam) in 2010? A. Distribution of data in a sample B. Sampling distribution C. Population distribution

A. Population distribution A population distribution is the distribution of values of a variable measured on individuals in a population. In this case it is the distribution of the number of friends of all Facebook pages.

Which of the following is the statistical name for the distribution of the number of friends of all Facebook pages? A. Population distribution B. Distribution of data in a sample C. Sampling distribution

population distribution. A population distribution is the distribution of values of a variable measured on individuals in a population. In this case it is the distribution of the weight of all adult female California sea lions.

Which of the following is the statistical name for the distribution of the weight of all adult female California sea lions?

Eric's. The law of large numbers states that as the sample size increases, the sample mean gets closer and closer to the population mean. So, we expect the sample mean from a sample of size 200 to be closer to μ = 10 than the sample mean from a sample of size 30.

Suppose Eric takes a random sample of size 200 from a population with mean μ = 10 and suppose Evelyn takes a random sample of size 30 from that same population. Whose sample do we expect to have a mean closer to the population mean of 10?

Z

Steps for Success -- Finding Normal Probabilities: 1. State the problem. 2. Draw a picture. 3. Compute _____. 4. Use table A. 5. Answer the question.

False. According to the central limit theorem, a sampling distribution becomes more Normal as the sample size increases. So, the shape of a sampling distribution is closer to Normal than the population distribution. This does not mean that the shape of the sampling distribution is approximately Normal.

True or false: The distributions of means of random samples are less Normal than the distribution of individual observations when the population distribution is not-Normal.

True. A population distribution is the distribution of values of a variable measured on individuals in a population.

True or false: The population distribution describes individuals in a population.

True. The variable of a sampling distribution is a statistic. For example, a sampling distribution could be the distribution of all the means from all possible samples from a population distribution.

True or false: The variable of a sampling distribution is a statistic.

decreases

As the sample size increases, the variability of the sample mean of the observed values _____________.

true mean

Recall that with a larger sample size, the sample mean gets increasingly closer to the ______ ________; therefore, having a narrower range of possible values and smaller variability.

Law of large numbers

As we sample more and more individuals from the population, we expect the *sample mean* to get *closer* to the *population mean*.

random sample

Do we have a random sample? Was it a randomized experiment? If neither, is the sample representative of the population?

population: sample ratio

Is the population at least 20 times larger than the sample?

central limit theorem

There are a few conditions that we will check to make sure that we can use the __________ _______ ____________: 1. Random sample 2. Population: Sample ratio 3. Large enough sample

False. A parameter is a number that describes the entire population.

True or false: A statistic is a number that describes the entire population.

The standard deviations for the sample mean decrease, and we're further in the tail of the distributions. It is becoming decreasingly likely to find a mean height about 67" as the sample size increases.

Why does the probability of interest decrease as the sample size increases?

statistic

a number that can be computed from the sample data without making any use of unknown parameters. In practice, we often use this to estimate an unknown parameter... Example: the average number of text messages sent yesterday by a random sample of OSU juniors.

parameter

a number that describes a characteristic of the population... Often the value of a parameter is unknown because we cannot examine the entire population... Example: the average number of text messages sent yesterday by all Ohio State students.

True. According to the law of large numbers, as the sample size increases, the sample mean gets closer and closer to μ. Thus, the law of large numbers guarantees the house its 5% profit resulting in a loss for a gambler in the long run.

A gambler at a casino decided to play a certain card game. The probability that the gambler will win money at the game is 0.45. The probability that he will lose money is 0.55. True or false: According to the law of large numbers, the longer the gambler plays, the more likely he/she is to end up with negative average winnings.

False. According to the law of large numbers, as the sample size increases, the sample mean gets closer and closer to μ. Thus, the law of large numbers guarantees the house a profit resulting in a loss for a gambler in the long run.

A gambler at a casino decided to play a certain card game. The probability that the gambler will win money at the game is 0.45. The probability that he will lose money is 0.55. True or false: According to the law of large numbers, the longer the gambler plays, the more likely he/she is to end up with positive average winnings.

representative

A primary tenet was that if we obtained a ____________________ sample of some phenomenon, then the rules of probability would allow us to generalize information from our random sample to make a statement about our population of interest.

sampling A statistic is the variable of a sampling distribution. For example, a sampling distribution could be the distribution of all the means from all possible samples from a population distribution.

A statistic is the variable of a __________ distribution.

central limit theorem

States that for large n, the sampling distribution of the sample mean x̄ is approximately Normal for any population with mean µ and finite standard deviation σ. That is, averages are more Normal than individual observations.

2.0 The standard deviation of the sampling distribution of x¯ equals σ/square root n. So, the standard deviation of the sample means from all possible samples equals the population standard deviation divided by the square root of the sample size.

For random samples of size 16 from a population with mean μ = 48 and standard deviation σ = 8, what is the standard deviation of the sampling distribution of x¯?

If individual observations have the N(μ, σ) distribution, then probability theory tells us that the sample mean of an SRS of size n has the N(μ, σ/square root n) distribution.

If individual observations have the N(μ, σ) distribution, then probability theory tells us that the sample mean of an SRS of size n has the N(μ, σ/square root n) distribution. Copy.

less variable

Note that the standard deviation for the sampling distribution of sample mean is population mean/square root n. This tells us averages are ______ ___________ than individual observations... Because the square root of n is in the denominator, we also know that the results of large samples are less variable than the results of small samples.

unbiased estimator

Sample mean in an ___________ ____________ of population mean.

Susan's. The law of large numbers states that as the sample size increases, the sample mean gets closer and closer to the population mean. So, we expect the sample mean from a sample of size 390 to be closer to μ = 48 than the sample mean from a sample of size 35.

Suppose Jacob takes a random sample of size 35 from a population with mean μ = 48 and suppose Susan takes a random sample of size 390 from that same population. Whose sample do we expect to have a mean closer to the population mean of 48?

Suppose that sample mean is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the sampling distribution of sample mean has mean μ and standard deviation σ/square root n.

Suppose that sample mean is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the sampling distribution of sample mean has mean μ and standard deviation σ/square root n. Copy.

Yes. The law of large numbers states that as the sample size increases, the sample mean gets closer and closer to the population mean. So, we expect the sample mean from a sample of size 1000 to be closer to μ = 21 than the sample mean from a sample of size 100.

Suppose we take a random sample of 100 students who took the ACT test in 2010 and get a sample mean of x¯ = 23.6. If we take a different sample of size 1000, do we expect the sample be closer to the population mean of μ = 21?

No. The law of large numbers states that as the sample size increases, the sample mean gets closer and closer to the population mean. So, we expect the sample mean from a sample of size 1000 to be closer to μ = 21 than the sample mean from a sample of size 100.

Suppose we take a random sample of 1000 students who took the ACT test in 2010 and get a sample mean of x¯ = 23.6. If we take a different sample of size 100, do we expect the sample mean from that sample be closer to μ = 21?

individuals

The *population distribution* is about the _______________ in the population, while the *sampling distribution* is about the values of the statistic calculated from the samples.

sampling distribution

The _____________ ______________ of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

population distribution

The ______________ _____________ of a variable is the distribution of the values of the variable among all the individuals in the population.

True. The shape of the sampling distribution of x¯ gets closer and closer to Normal as the sample size increases. So, with a sample of size 5, the shape of the sampling distribution will be less skewed than the population, but not quite Normal.

The amount of caffeine consumed per day by children aged eight to twelve years old has a right skewed distribution with mean μ = 110 mg and standard deviation σ = 30 mg. True or false: The shape of the sampling distribution of x¯ for samples of size n = 5 will be less skewed than the population, but not approximately Normal either.

statistic. It is a statistic because it is a number that can be computed from the sample of 97 children.

The average amount of caffeine consumed per day for a sample of 97 eight to twelve year-olds was 78 mg. The number 78 is a __________.

statistic. It is a statistic because it is a number that can be computed from the sample of 104 children.

The average number of hours of sleep per night was 9.46 hours for a sample of 104 five to seven year-old children. The number 9.46 is a __________.

Normal

The central limit theorem allows us to use Normal probability calculations to answer questions about sample means from many observations - even when the population distribution is not ____________.

unknown

The law of large numbers says, "sample enough individuals and the statistic sample mean will approach the ____________ parameter population mean."

The probability that the sample mean is exactly equal to the population mean is 0. However, we expect the sample mean to be somewhere near the population mean. The sample mean is an unbiased estimator of the population mean -- we don't expect it to over or underestimate the population mean.

The probability that the sample mean is exactly equal to the population mean is 0. However, we expect the sample mean to be somewhere near the population mean. The sample mean is an unbiased estimator of the population mean -- we don't expect it to over or underestimate the population mean. Copy.

equal to The mean of the sampling distribution of x¯ is always equal to the mean of the population distribution.

The shape of a population distribution is uniform with mean μ = 2 and standard deviation σ = 0.4. The sampling distribution of x¯ is created from the sample means from all possible samples of size 64. How does the mean of this sampling distribution compare with the mean of the population distribution? The mean of the sampling distribution is _______ _____ the mean of the population distribution.

normal instead of uniform like According to the central limit theorem, the shape of a sampling distribution gets closer to Normal as the sample size increases. With a sample of size 64, the sampling distribution of x¯ is approximately Normal instead of uniform like the population distribution.

The shape of a population distribution is uniform with mean μ = 2 and standard deviation σ = 0.4. The sampling distribution of x¯ is created from the sample means from all possible samples of size 64. The shape of this sampling distribution is ____________ the shape of its population distribution.

less than The standard deviation of the sampling distribution of x¯ equals σ/square root n = 0.4/square root 64 = 0.05. This is less than the standard deviation of the population distribution, which is σ = 0.4.

The shape of a population distribution is uniform with mean μ = 2 and standard deviation σ = 0.4. The sampling distribution of x¯ is created from the sample means from all possible samples of size 64. The standard deviation of the sampling distribution is ______ _____ the standard deviation of the population distribution.

Normal. If individual observations have a Normal distribution, then the sample mean x¯ of an SRS of size n is also Normal.

The shape of the sampling distribution of x¯ created from random samples from a Normal population is ______________.