Exam 2

NORM.DIST

??

NORM.INV

??

P-Value

A probability that provides a measure of the evidence against the null hypothesis provided by a sample

Sample

A subset of the population

Interval Estimate

A/An _________ can be computed by adding and subtracting a margin of error to the point estimate

Larger

Better precision/Higher confidence means _______ sample sizes

Yes, depending on the structure of the population

Can a sample size of 5 be sufficient?

Method used when the population is not normally distributed; the sampling distribution becomes normal with a large n (n=30, maybe 50 when population is highly skewed or there are many outliers)

Central Limit Theorem

CONFIDENCE.T(alpha,standard_dev,size)

Confidence Function Syntax:

1. Equal to n-1 2. Logic: n pieces of information are used in developing 3. But we know that -If we know xi ... xn-1 we can always calculate xn -So, only n -1 of the values are independent

Degrees of Freedom for the t Distribution:

The size of each sample

Does n equal the number of samples of the size of each sample?

NORM.S. DIST NORM.S. INV NORM.DIST NORM.INV CONFIDENCE.T T.INV T.DIST T.DIST.RT T.DIST.2T

Excel Functions Used:

NORM.S. DIST

Excel formula that computes the cumulative probability given a z value

NORM.S. INV

Excel formula that computes the z value given a cumulative probability

No. The standard deviation (s) of each sample will be different. However, in class 2 (slide 13) we demonstrated that s is an unbiased estimator of standard deviation. So, on average the intervals will converge to standard deviation (a constant)

For the standard deviation known case we pointed out that the confidence interval for every xi will be equidistant. Is this also true for the case when standard deviation is unknown?

Abraham De Moivre

French mathematician who derives the normal distribution in 1733

The interval cannot include the true mean; Reject it

How do we know that all x samples in the tail of the sampling distribution will not include the mean in their confidence interval?

The interval estimate is 2 margin of errors, one added and one subtracted

How does the margin of error compare to the interval estimate?

They are the exact same!

How does the standard error of the mean compare to the standard deviation of the sampling distribution?

As n increases, the standard deviation of the sample distribution shrinks/becomes tighter

How is a statistical inference helped by a larger sample size?

Sample Standard Deviation (s)

If an estimate of the population standard deviation (s) cannot be developed prior to sampling, we use the ________ to estimate the standard deviation

T distribution

If an estimate of the population standard deviation (s) cannot be developed prior to sampling, we use the sample standard deviation s to estimate the standard deviation. In this case, the interval estimate for m is based on the _______

1. This is the distribution of the statistic 2. For each sample that we take, we can make an inference about the quality of that sample based on the distribution statistic 3. Note that the distribution of the statistic is much "tighter" than the distribution of the population -Each sample point is a statistic (average, in this class) -The standard deviation of the sample measured the variability of the averages (in this case)

If we take many samples, the samples will have a distribution:

1. The population standard deviation (o), or 2. The sample standard deviation (s)

In order to develop an interval estimate of a population mean, the margin of error must be computed using either:

- The t distribution is used when the same sample is used to estimate x and s. Those estimates cant be made until the sample (for which we are determining n) is taken - The t distribution is a function of n

In this class we used the normal distribution (z) to estimate the sample size to determine an interval estimate for the mean when the standard deviation is unknown. Why didn't we use the t distribution?

Infinitie

Infinite or Finite: Boxes of cereal off a production line

Infinite

Infinite or Finite: Cars crossing the Golden Gate Bridge on a particular day

Finitie

Infinite or Finite: Licensed drivers in the state of New York

Infinite

Infinite or Finite: Sample of orders processed by Amazon

Finite

Infinite or Finite: Students in this course

Unknown

Is it more likely to have a known or unknown standard deviation?

Frame

List of elements that the sample will be drawn (example: list of the registered voters in Texas)

1. Height, weights, scientific measurements 2. Results from samples--statistical inference

Normal Distributions Is Very Common:

1. The mean can have any value 2. It is asymptotic (infinite)

Normal Probability Distribution

Parameters

Numerical characteristics of a population (mean, standard deviation, p)

Sample Statistic

Numerical characteristics of a sample (x, s, p)

We really do not know it

Particularly speaking, when do we know the standard deviation?

Sampled Population

Population from which the sample is drawn (example: registered voters in Texas)

When p is extreme or there is a small sample: np is greater than or equal to 5 n(1-p) is greater than or equal to 5

Practically speaking, when should we be concerned about using the normal probability distribution to approximate a binomial distribution?

1. If the population follows a normal distribution the confidence (ex. - 95%) associated with the intervals is exact. Otherwise, it is an approximation. 2. In most applications, a sample size of n = 30 is adequate 3. If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended 4. If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 may be sufficient 5. If the population is believed to be at least approximately normal, a sample size of less than 15 can be used

Sample Size Requirements for Estimating the Population Mean (this is the same if the standard deviation is known or unknown):

A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample has the same probability of being selected

Sampling from a finite population:

A random sample of size n from an infinite population is a sample selected such that each element selected: 1. Comes from the same population, and 2. Is selected independently

Sampling from an infinite population:

You could use either

Suppose standard deviation is unknown, but you have a very large sample (n=2,000). Should you use a t or z distribution to determine your confidence intervals?

That might not be the true mean

Suppose the mean of your sample of a population is 3 standard deviations away from the mean of the population. You decide to sample the population again and your sample is 3.1 standard deviations from the mean. What is your inference?

As n increases, we will know if it is a normal distribution

Suppose we take several samples from a population with an unknown distribution. What will the distribution of the averages of those samples?

The sample wasn't representative of the true population

Suppose we want to project whether a local ballot initiative will pass. We go to several neighborhood "town hall meetings" around Houston and sample opinions from those in attendance. We determine that the ballot is unlikely to pass. But on election day the initiative passes with a strong majority. What happened?

No, you don't know if the service is fast or slow

Suppose you are a server and your service times are uniformly distributed. Will all of your customers be equally happy?

The confidence interval will be larger with a smaller n

Suppose you know the standard deviation but you use the t distribution (instead of the z distribution) to determine your confidence interval. What effect will this have on your confidence interval?

Whatever the value of alpha is

Suppose you reject the null (Ho) hypothesis, what is your potential type 1 error?

T.DIST(x,deg_freedom, cumulative)

T.DIST Function Syntax:

T.INV(probability,deg_freedom)

T.INT Function Syntax:

True

The Microsoft Excel function NORM.S.DIST evaluates the standard normal distribution

1. Is the most important distribution for describing a continuous random variable 2. Is widely used in statistical inference 3. Has been use in a wide variety of applications

The Norma Probability Distribution:

Critical Value Approach

The _______ approach uses the z-value that provides an area of a in the tail of the standard normal distribution

Null Hypothesis (H0)

The _______ hypothesis is a tentative assumption about a population parameter; it is meant to be challenged

Alternative Hypothesis (Ha)

The _______ is the complement of the null hypothesis

Population

The collection of all elements of interest (example: registered voters)

Element

The entity on which data are collected (example: person, fund, company)

Null Hypothesis

The equality part of the hypothesis always appears in the _______ hypothesis

The sampling distribution of x

The probability distribution of all possible values of the sample mean

The sampling distribution of p

The probability distribution of all possible values of the sample proportion

The level of significance (alpha)

The probability of making a type I error when the null hypothesis is true as an equality

True

True of False: If the population is normally distributed, the sampling distribution will be normal

True

True or False: =CONFIDENCE.NORM(0.05,5,30) The above Microsoft Excel function will return an approximate value of 1.79

True

True or False: =NORM.INV(0.7,15,4) The above Microsoft Excel function will return an approximate value of 17.1

False

True or False: =T.DIST(a,b,c) The value of "c" in the above Microsoft Excel function is equal to the degrees of freedom

True

True or False: A 100% confidence interval will be infinite

True

True or False: A higher level of confidence for an estimate of a population parameter will require a larger sample size (everything else remaining constant)

False

True or False: A hypothesis test for an interval estimate that includes the hypothesized parameter value should be rejected

False

True or False: A p-value greater than α will always result in a rejection of the null hypothesis

True

True or False: A point estimator cannot be expected to provide the exact value of the population parameter

True

True or False: As the sample size increases the t probability distribution approaches the normal probability distribution

True

True or False: If 100 samples are taken of a process, a 90% confidence interval for each sample will include the population mean 90 out of a hundred times (on average)

True

True or False: If a population is not normally distributed, the distribution of the means of samples taken from that distribution will be normally distributed when the samples are larger than 5

False

True or False: If a variable is continuously and uniformly distributed between 10 and 25, the variance of the variable is equal to 15

True

True or False: If a variable is continuously and uniformly distributed between 10 and 40, the expected value of the variable is equal to 25

True

True or False: If you do not know the population standard deviation, but know the maximum (2,400) and minimum (400) values in the population, a reasonable estimate of the population standard deviation would be 500

False

True or False: If you have a research hypothesis, you should include it in your model as the null hypothesis

True

True or False: Inferences are not always correct

True

True or False: It is important to make sure that the frame captures the sampled population and the sampled population is representative of the population of interest

False

True or False: It would be reasonable to use the uniform probability distribution to model the arrival times of customers at a restaurant

True

True or False: It would be reasonable to use the uniform probability distribution to model the arrival times of customers at a restaurant

True

True or False: On average, you would expect to incur a type 1 error when rejecting the null hypothesis at a rate proportional to alpha

False

True or False: Rejecting a null hypothesis when the null hypothesis is true is a type II error

True

True or False: The area in the tail of the sampling distribution for a one-tailed hypothesis test is equal to alpha (α)

True

True or False: The area within 1.8 standard deviations (in either direction) of the mean of the standard normal probability distribution is closer to 0.92 than to 0.82

False

True or False: The area within one standard deviation (in either direction) of the mean of the standard normal probability distribution is closer to .90 than to .70

True

True or False: The daily products off a high-volume assembly line is an example of an infinite population

True

True or False: The expected value for p-bar equals p

False

True or False: The expected value of the continuous uniform probability distribution is equal to the sum of the end points of the density function

False

True or False: The number of different independent samples of 10 in a finite population of 12 is equal to 4

True

True or False: The population mean is assumed to be a fixed parameter

True

True or False: The population standard deviation is rarely known exactly, but often a good estimate can be obtained based on historical data or other information

True

True or False: The probability of a single point in a continuous probability distribution equals zero

True

True or False: The sampling distribution of p-bar is the probability distribution of all possible values of the sample proportion (p-bar)

True

True or False: The standard deviation of the standard normal probability distribution is equal to 1.0

False

True or False: The standard deviation of the standard normal probability distribution is equal to zero

True

True or False: The standard error of the mean is another name for the standard deviation of the sampling distribution

True

True or False: The standard error of the proportion equals the standard deviation of the sampling distribution of p-bar

True

True or False: The true distribution for a proportion is a discrete binomial distribution

True

True or False: The width of an interval estimate of a population parameter is equal to two times the desired margin of error

False

True or False: Values of p-bar are unbiased estimates of the p-value

True

True or False: a p-value less than alpha will always result in a rejection of the null hypothesis

We can evaluate our inference based on a normal distribution, instead of a binomial distribution

We showed that the distribution of the population of a binomially distributed x can be approximated by a normal distribution. How does this result affect our analysis of the sample statistic p?

Alpha should be very small

What changes should you make if the cost of potential type 1 errors is high?

Every sample point is x bar, the average of that sample

What do the individual sample points equal in the previous sampling distributions?

1

What does the cumulative area under the standard normal curve equal?

How close the point estimate is to the value of the parameter

What does the interval estimate provides information about?

Margin of Error- When E is smaller, n will be bigger Level of Confidence- When there is a higher level of confidence, n will be bigger Variability of Population- More precision means a larger sample size

What effect does each of the following have on sample size requirements? Margin of error (E)? Level of confidence? Variability of population?

Range/4

What equation gives a rough approximation of the standard deviation?

T.DIST.(t,deg_freedom,TRUE)

What function is used for a lower-tailed test when the population standard deviation is unknown?

T.DIST.2T(t, deg_freedom)

What function is used for a two-failed test when the population standard deviation is unknown?

T.DIST.RT(t,deg_freedom)

What function is used for an upper-tail test when the population standard deviation is unknown?

You need an estimation of the standard deviation and you will use a t-table instead of a z-table

What is the difference between testing hypotheses when the standard deviation is known and when the standard deviation is unknown?

p*

What is the planning value variable?

1. Develop the null and alternative hypotheses 2. Specify the desired level of significance 3. Collect data and compute the test statistic 4. Evaluate the hypothesis (p-value or critical value)

What is the procedure for testing a hypothesis?

There will always be an error, but the interval estimate tells more

What is the purpose of using interval estimate (as opposed to point estimates) of population parameters?

Have a bigger n for a smaller confidence interval

What is the width of the confidence interval is impractical (very wide)? What should we do?

Negative (-) Infinity

What is the z-value for a cumulative probability of 0?

Positive (+) Infinity

What is the z-value from a cumulative probability?

.5

What value results in the largest n (when using a conservative estimate)?

Infinity

What will the 100% confidence interval equal?

Z-Values/Table

When estimating the population mean, we use _______ to determine the interval estimate when know the standard deviation

Sample Mean

Which mean varies with each sample?

Level 1 Error

Which type of error does the level of significance (alpha) control?

The normal distribution (z) is an reasonable approximation of the binomial

Why do we use the normal distribution (z) to test hypothesis about the population proportion (p)?

No, you'll always come to the same conclusion/get the same answer

Will the p-value approach ever result in a different conclusion than the critical value approach?

Smaller

_______ p-values indicate more evidence to reject Ho