Midterm #2: Review Guide

If you increase the sample size, an all other values remain the same, the width of confidence intervals will increase. True or false?

(*Sample MT, #1*) If you *increase sample size*, standard error *decreases*, which means the width (confidence interval) *decreases*.

area score (*3*)

(Discussion 3, #1) 1.) area under the curve *between* the mean and z score. 2.) Always *positive*. (Can't have a negative probability!) 3.) Area of curve is equal it '1'.

z-score (*2*)

(Discussion 3, #1) 1.) z-score: area under the standard deviation *above* or *below* the mean. 2.) Can be *positive* or *negative*.

Suppose you are asked to find the 20th percentile and the 80th percentile for a set of scores. These two problems are solved almost exactly the same. Draw the diagram for each and discuss the part of the solution that would be different in finding the requested probabilities.

(Discussion 3, #2) P20 = Area Score (0.20) | z score = negative (Because you're using *negative* table.) P80 = Area Score (0.80) | z score = positive (Because you're using *positive* side of z table.)

Q1, Q2, Q3, Q4. Find the z value closest.

(Discussion 3, #7 | Practice Midterm, #7) Q1 = 25. z value with area closest to .25 = x = *-0.67* (.2514 is CLOSER than .2483.) Q2 = 50. z value with area closest to .50 = x = "0.00" (.5000 is CLOSER than .5040.) Q3 = 75. z value with area closest to .75 = x = *0.67* (.7486 is CLOSER than .7517.) Q4 = 100. z value with area closest to 100 = = '3.50' and up. (.9999 is the only value to choose.)

Best point estimate for the *true population proportion* (p(hat)? (*2*)

(Discussion 5, #2) Sample proportion. This is p(hat) = r/n. 1) P is unbiased (does not overestimate or underestimate probability). 2) p(hat) is the most consistent (has least variation of al measures of tendency.

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

(Homework 6, #14) areas, probability, and relative frequencies

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

(Homework 6, #15) Uniform. The graph of a uniform distribution is rectangular because the values are spread evenly over a range. That means every value in the range is equally likely.

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

(Homework 6, #16) Finding the area of the shaded region representing that probability.

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

(Homework 6, #29) a data value associated with an area of 0.85 to its left.

The expression z'alpha' denotes the z score with an area of 'alpha' :

(Homework 6, #3) To its right.

___ for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values.

(Homework 7, #25) Degrees of Freedom

A critical value, zα, denotes the _

(Homework 8, #11) Z score with an area of α to its right.

The Student t distribution has the same general bell shape as the standard normal distribution. True or False?

(Homework 8, #12) False. The mean is t=0, but the standard deviation of the Student t distribution varies with the sample size and is greater than 1.

The sample mean, x bar is greater than 30. True or False?

(Homework 8, #13) False. n > 30.

The ___ hypothesis is a statement that the value of a population parameter is equal to some claimed value.

(Homework 8, #2) Null.

The ___________ is a value used in making a decision about the null hypothesis and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.

(Homework 8, #3) Test Statistic. It is found by converting the sample statistic (p hat, x bar, or s) to a score (such as z, t, or chiy squared2) with the assumption that the null hypothesis is true.

The P-value separates the critical region from the values that do not lead to rejection of the null hypothesis. True or False?

(Homework 8, #4) False. The p-value is an area.

The inequality symbol in the alternative hypothesis points away from the critical region. True or False?

(Homework 8, #5) False. It points in the direction of the critical region.

Conditions of a *Normal* Distribution (*6*)

(Lecture #12, 2-6-19) 1) Bell shaped curve. 2) Symmetric 3) Centered around the mean 4) Mean = media = mode 5) Governed by the parameters, 'mu' and 'sigma' 6) 'N' is distributed Normal (*'mu'*, *'sigma'*)

Conditions of a *Standard Normal* Distribution (*5*)

(Lecture #12, 2-6-19) 1) Bell shaped curve. 2) Symmetric 3) The mean 'mu' is 0. The standard deviation 'sigma' is 1. 4) Centered around '0'. 5) 'X' is distributed Normal (0,1).

Continuous Random Variable

(Lecture #12, 2-6-19) The RV (random variable) is a continuous random variable if X consists of all real numbers.

(The z-score) represents the _.. This value is represented in our _.

(Lecture #12, 2-6-19) (Normal Probability Tables: Z-Table) # of standard deviations observations (x) are *above* or *below* the mean. Standard normal distribution (see notes).

- When 'z' is *below* '0', it is _. - When 'z' is *above* '0' it is _.

(Lecture #12, 2-6-19) (Normal Probability Tables: Z-Table) Negative. Positive.

'Z' is _ while 'z' is _.

(Lecture #12, 2-6-19) (Normal Probability Tables: Z-Table) Z: Random Variable, z: Observed Value (Normal Probability Tables: Z-Table)

Equation for Uniform Distribution

(Lecture #13, 2-8-19) 1 / b - a

Uniform Distribution - Continuous (*3*)

(Lecture #13, 2-8-19) 1) Rectangular Distribution 2) Parameters a (minimum) and b (maximum) 3.) Probability Distribution with constant probability between a and b.

If you are not told explicitly, the confidence interval you will use is:

*.95* (It is the most common!)

The CLT can be applied if (*4*)

(Lecture #13, 2-8-19) 1.) X(bar) is distributed normally ('mu", 'sigma' / square root of 'n'). (*AND/OR*) 2.) n > 30, for non-normal distributions 3.) n: sample size. 4.) 'mu' mean of original population 5.) 'sigma' is standard deviation of original population.

State the Central Limit Theorem (CLT)

(Lecture #13, 2-8-19) For normal and non-normal populations, the distributions of the sample mean, x(bar), will approximately follow a normal distribution with a mean ('mu' = x(bar) and standard deviation = ('sigma' / square root of 'n')

By Empirical Rule, we know that:

(Lecture #14, 2-11-19) +'sigma', -'sigma' = 'More than one' = 0.68. + *2*'sigma', -*2"'sigma' = 'More than two' = 0.95 +*3* 'sigma', - *3*'sigma = 'More than three = 0.997 (We are (95) percent confident that the true falls between M - *2*sigma and *M + *2* sigma.)

Define a Confidence Interval.

(Lecture #14, 2-11-19) A 100 (1 - 'alpha')% confidence interval for the population mean 'mu' is a symmetric interval about your sample or point estimate of 'mu', which is x(bar).

Z 'alpha'/2 =

(Lecture #14, 2-11-19) Critical Value

1 - 'alpha' =

(Lecture #14, 2-11-19) Level of Confidence

E =

(Lecture #14, 2-11-19) Margin of Error

'alpha' =

(Lecture #14, 2-11-19) Significance Level

Define a point estimate.

(Lecture #14, 2-11-19) Single value (or point) used to approximate population parameter.

*Descriptive* Statistics vs. *Inferential* Statistics

(Lecture #14, 2-11-19) (Confidence Intervals) *Descriptive* Statistics: summarizes characteristics of inferences about the known *population data*. *Inferential* Statistics: Uses *sample data* to make inferences about the population.

What is the best point estimate for *population mean* (μ)? (*2*)

(Lecture #14, 2-11-19) (Discussion 4, #6) Sample mean. This is *(xbar)*. 1.) x(bar) is unbiased (does not systematically underestimate or overestimate the true population mean). 2.) x bar targets the population mean.

What happens when 'sigma' is not given?

(Lecture #15, 2-13-19) Population standard deviation is not given, we use 's' (sample standard deviation). We use 's' in place of 'sigma' which introduces additional error, casing the spread of the curve to increase. Thus, the distribution is no longer normal and we will use the *student's t distribution*.

Based on only one interval, the probability that it contains 'mu' is:

(Lecture #15, 2-13-19) '0 or 1'.

If we sample repeatedly, say 100 times, and calculate the interval for each sample, then a 90% confidence means that:

(Lecture #15, 2-13-19) 90 intervals will contain the *true mean* and 10 *will not*.

Why do we use the word 'confident'?

(Lecture #15, 2-13-19) Because the 'lower bounds and upper bounds' (parameter) are NOT fixed values. They will change based on your sample. DO NOT use probability!

If I increase my level of confidence, what happens to the interval?

(Lecture #15, 2-13-19) It will *widen* (not narrow!).

Student t Distribution (*7*)

(Lecture #15, 2-13-19) (Confidence Intervals) 1) Varies for different sample sizes (n) 2) Governed by degrees of freedom: n - 1. 3) Same shape as normal, but more variability. 4) It is wider and flatter. 5) mean: t = 0. 6) Standard deviation varies according to n but great than '1'. 7) As sample size increase, it looks more normal.

Confidence Interval for the true population mean, 'mu' (Z Table) (*3*)

(Lecture #16, 2-15-19) 1.) 'sigma' is *known*. 2.) X is distributed normally and/or 3.) n > 30.

Confidence Interval for the true population mean, 'mu' (T Table) (*3*)

(Lecture #16, 2-15-19) 1.) 'sigma' is *unknown* (but 's' is!) 2.) X is distributed normally and/or 3.) n > 30.

What are the five steps for a hypothesis test? (Will be provided for us if needed on exam.)

(Lecture #17, 2-20-19) 1.) Determine Ho and Ha. 2.) Calculate Test Statistic 3.) Calculate P Value 4.) State your decision 5.) State your conclusion in terms of the claim.

To conduct a hypothesis test, we must first determine (*2*)

(Lecture #17, 2-20-19) 1.) What population parameter you are interested in. 2.) Check to see necessary assumptions are met.

We could use the confidence interval to see if our parameter lies within the particular parameter we are looking for, or we can conduct a:

(Lecture #17, 2-20-19) Hypothesis Test

Alternative Hypothesis (Ha) (*Step *1*)

(Lecture #17, 2-20-19) Statement we adopt in the situation in which the evidence (data) is so strong that we reject Ho.

Interpretation: (*Step 2*)

(Lecture #17, 2-20-19) The estimated number of standard deviations away from the hypothetical value (Mo or po). If large (far away), then our data does not agree with Ho and we support Ha. (Hypothesis Testing)

We assume that Ho is (true/false) and test to see if there is enough evidence to support the alternative hypothesis. (*Step *1*)

(Lecture #17, 2-20-19) True.

If 'sigma' is unknown & mean (*Step 2*), use:

(Lecture #17, 2-20-19) Use population mean for *'t' test* statistic (equation). (Hypothesis Testing)

If 'sigma' is known & mean (*Step 2*), use:

(Lecture #17, 2-20-19) Use population mean for *'z'* test statistic (equation). (Hypothesis Testing)

If proportion (*Step 2*), use:

(Lecture #17, 2-20-19) Use population proportion for *'z' test* statistic (Hypothesis).

Suppose we want to collect sample data to estimate some population proportion. How many samples must be obtained?

(Lecture #17, 2-20-19) Use the margin of error, 'E'. If p(hat)/q(hat) are not given, then p(hat) = 0.5. (Hypothesis Testing)

In Hypothesis Testing, we are interested in determining if the population parameter ('mu', p, 'mu''1 - 'mu'2, 'p'1 - 'p' 2, is _ some particular value.

(Lecture #17, 2-20-19) greater than, equal to, greater than ,or less than some particular value.

Null Hypothesis (Ho) (*Step 1*)

(Lecture #17, 2-20-19) statement under investigation or being tested. It represents 'no effect'/'no difference'/ 'unchanged.'

If p value < 'alpha', then: If p-value is > 'alpha', then: (*Step 3*)

(Lecture #18, 2-22-19) < Reject Ho. > Fait to reject (FTR) Ho. (Remember: If p is low, null must go. If po is high, the null will fly.

State your decision (*Step 4*)

(Lecture #18, 2-22-19) Reject Ho? FTR Ho?

Compare the -pvalue to the _

(Lecture #18, 2-22-19) Significant Level, 'alpha' (0.01, 0.05, 0.10). Will be given if we need them. (Hypothesis Testing)

Smaller the p-value, (*Step 3*)

(Lecture #18, 2-22-19) Smaller the p-value, the less likely our data supports/agrees with Ho. (Hypothesis Testing)

Calculate the p-value (*Step 3*)

(Lecture #18, 2-22-19) The probability that you observe a value of the test statistic as extreme or more (in step 2) than the one from the sample, given Ho is true. (Hypothesis Testing)

Conditions for the true *proportion*, p (*3*):

(Lecture 16, 2-15-19) 1.) Let r (a.k.a 'x') be number of successes of 'n' trials. 2.) Sample proportion of successes is denoted p(hat) r/n. 3.) For the normal approximation of the binomial, we must have n*p(hat) > 5, and n*q(hat) > 5.

If you take large random samples over and over again from the same population, and make 95% confidence intervals from the population average, about 95% of the intervals should contain the population mean. True or False?

(Practice Midterm, #2) True. This is based on the definition of confidence intervals. Given Empirical Rule, we know ('mu' - 2'sigma', 'mu' + 2'sigma') = 0.95. Hence, we are 95% confident the true mean falls between 'mu' +/- 2'sigma'.

CLT guarantees that the population mean is normally distributed whenever sample size is sufficiently large. True or False?

(Practice Midterm, #3) False. It guarantees the distribution of sample means, NOT the sample, follow a normal distribution when 'n' is very large.

A p-value of .08 is more evidence against the null hypothesis than a p-value of .04. True or false?

(Practice Midterm, #4) False. a p-value of 0.08 is more evidence to support, NOT against, our null hypothesis than a p value of 0..04.

Certain test scores are normally distributed with a mean of 60 and standard deviation of 4. What are the values of the mean and standard deviation after all test scores have been standardized by converting them to z-scores?

(Quiz 3, #1) The mean is 0 and standard deviation is 1.

I can calculate probability for a Uniformly distributed random variable by calculating the area of a:

(Quiz 3, #10) Rectangle

When you use the standard normal table, area under the curve is always given to the _____ of the value.

(Quiz 3, #11) Left

Suppose you want to construct a confidence interval for the true population mean. You would use the Student's t Table to find the critical value when:

(Quiz 3, #12) 'sigma' is unknown but 's' is given.

Suppose you want to construct a confidence interval for the true population mean. You would use the standard normal table to find the critical value when:

(Quiz 3, #12) sigma is known

The Central Limit Theorem describes the distribution of the sample

(Quiz 3, #14) means

Sample Proportion is:

(Quiz 3, #15) p(hat) = r/n.

The interpretation of confidence intervals is based on the idea of repeated sampling. If I construct 10 intervals for the true population mean, then at 90% confidence, ____ intervals will contain the mean and ____ will not.

(Quiz 3, #16) (9, 1)

To construct a confidence interval for the mean or the proportion, we take the sample estimate and add/subtract E. For example, for the population mean we calculate x(bar) +/- E . Here, E represents the estimate:

(Quiz 3, #18) Margin of Error.

The Student's t distribution is based on degrees of freedom: d f = n − 1. As the sample size increases, the distribution looks:

(Quiz 3, #19) More normal.

If selecting samples of size n = 10 from a *population* with a known mean and standard deviation, what requirement, if any, must be satisfied in order to assume that the distribution of the sample means is a normal distribution?

(Quiz 3, #2) The population must have a normal distribution.

z scores cannot be negative.

(Quiz 3, #20) False! Z Scores can be positive or negative.

If selecting samples of size n > 30 from a population with a known mean and standard deviation, what requirement, if any, must be satisfied in order to assume that the distribution of the sample means is a normal distribution?

(Quiz 3, #3) None; the distribution of the sample means will be approximately normal.

What is x(bar)?

(Quiz 3, #4) A random variable

The Student's t distribution is flatter and wider than the Normal Distribution. True or False?

(Quiz 3, #7) False.

Tina's score on her midterm exam was at the 50th percentile. The grades were normally distributed. The exam average was a 78 and the standard deviation was 6. What was Tina's score on the exam?

(Quiz 3, #8) 78.

A z-score of 2.3 on a normally distributed value corresponds to a percentile rank of:

(Quiz 3, #9) .9893 = 98.93

When interpreting a confidence interval, it is appropriate to use the term probability.

(Quiz 3, Question #17) False.

State your conclusion in terms of the claim (*Step 5*)

Identify claim from Step 1. "At the 'alpha'% significance or with (1-'alpha')% confidence, true _ is or is not significant evidence to support the claim that _." (Hypothesis Testing)

'alpha' (*Step 3*)

Probability of Type 1 Error P (Reject Ho / Ho True)

Always __ if degrees of freedom (df) are not listed on the time.

Round down!

According to the CLT, the mean of all sample means is_

the population mean 'mu'.

Z-Score formula GIVEN sample mean.

z = 'x'(bar) - 'mu' / 'sigma' / square root of n.

Find the standard deviation given 'Q' formula;

z = (x- 'mu') / 'sigma'. x = data point given. 'mu' = mean, also given. Find 'z' (corresponding 'z' score from table.)

Z score formula NOT given sample mean:

z = x - m / 'sigma'