Statistics Test #3

Power

(Power is the probability of rejecting H0 when H0 is false. ) The probability of rejecting a false null hypothesis; computed as 1 - beta. Increase power by increasing sample size or increasing alpha. (probability of making a CORRECT decision)

Confidence intervals: If CI excludes the value, data provides evidence to reject H0

(READ OTHER SIDE)

Confidence intervals: If CI includes the value, data provides support for H0

(READ OTHER SIDE)

Standard deviation of a sample

(s) A measure of the variability (spread) of the data in a sample.

Standard error of x-bar.

(s/ sqrt n) An estimate of the variability of the sampling distribution of x-bar; estimates the standard deviation of the sampling distribution of x-bar.

Standard deviation of x-bar

(sigma/sqrt n) A measure of the variability (spread) fo the sampling distribution of x-bar.

Under what conditions do confidence intervals work/when are they valid?

1. Random sampling 2. sampling distribution of xbar is normal 3. sigma is known

Four Steps for Tests of Significance

1. State (Specify claim about parameter of Interest) 2. Plan (Choose procedure, specify H0, Ha, alpha) 3. Solve (Collect data, check conditions, calculate test statistic and p-value) 4. Conclude (compare p-value to alpha, interpret test results)

Degrees of freedom

A characteristic of th t-distribution (e.g. n-1 for a one-sample t); a measure of the amount of information available for estimating sigma using s.

Practical importance

A difference between the observd statistic and the claimed parameter value that is large enough to be worth reporting. To assess practical importance, look at the numerator of the test statistic and ask, Is it worth anthing? If yes, then results are also of practical importance (Note: do not assess practical importance unless results are statistically significant)

t distribution

A distribution specified by degrees of freedom used to model test statistics for the one-sample t test, et. where sigma ('s) is (are) unknown. Also used to obtain confidence intervals and p-values for t-procedures

Test statistic

A number that summarizes the data for a tst of significance; usually used to obtain P-value.

One-sided or one-tailed test

A test where the alternative hypothesis contains either "<" or ">". The left-tailed test will have a "<" in the alternative hypothesis. The right-tailed test will have a ">" in the alternative hypothesis.

Standard error of a statistic

An estimate fo teh standard deviation of the sampling distribtuioin of the statistic; in other words, it is a measure of the variability of the statistic. Note: The denominators of most test statistics are called standard errors.

One-sample t test

An inferential statistical procedure that uses the mean for one sample of data for either estimating the mean of the population or testing whether the mean of the population equals some claimed value.

Which is more important of the t* procedure conditions?

Conditions: Normality or Randomness.... Randomness is the most important.

Statistically significant is equivalent to all of the following except one. Which one is NOT equivalent? a) P-value < α. b) The difference between the observed value of the statistic and the value of the parameter as given in H0 is too large to attribute to just chance variation. c) The probability of obtaining a sample statistic as extreme or more extreme than actually observed if H0 were true is too small for us to believe that H0 is correct. d) The observed statistic is inconsistent with the null hypothesis. e) The difference between an observed statistic and the true parameter value is due to chance variation.

E

Matched pairs

Either two measurements are taken on each individual such as pre and post OR two individuals are matched by a third variable (different from the explanatory variable and the response variable) such as identical twins.

T or F: For fixed α, increasing sample size increases β.

False

T or F: P-value is the probability that the null hypothesis is true.

False

T or F: Power is the probability that Ha is believed when H0 is true.

False

T or F: α is found in the fail to reject H0 region under the curve of the sampling distribution defined by H0.

False

True or False: A 95% confidence interval can be used to perform a one-sided test on H0: μ = μ0 versus Ha: μ > μ0 at α = 0.05.

False

True or False: If the probability of obtaining our sample data, assuming the null hypothesis were true, is large, we have enough evidence to accept the null hypothesis.

False

True or False: Statistical hypothesis testing is defined as assessing evidence provided by the data in favor of or against some claim about the sample.

False

True or False: The p-value is the probability that the null hypothesis is true.

False

True or False: When the population standard deviation, σ, is unknown, we cannot compute a confidence interval.

False

Increasing the sample size will _____ the accuracy of the point estimator. (Decrease, increase, not change)

Increase

Multiple Comparisons

Performing two or more tests of significance on the same data set. This inflates the overal alpha (probability of making a type I error) for the tests. (the more comparisons performed, the greater the chances of falsely rejecting at least one true null hypothesis.)

Which hypothesis is the researcher trying to prove generally?

The alternative hypothesis

Reject H0

The appropriate statistical conclusion when P-value < alpha.

Fail to reject H0

The appropriate statistical conclusion when the P-value is greater than alpha.

Conditions

The basic premises for inferential procedures. If the conditions are not met, the results may not be valid.

Null Hypothesis

The hypothesis of not difference or no change. This is the hypothesis that the researcher assumes to be true until sample results indicate otherwise. Generally it is the hypothesis that the researcher wants to disprove. (Note: Interpretations of P-value and statistically significnt need to say something about--if H0 is true in order to be correct.)

Matched pair t test

The hypothesis testing method for matched pairs data. The typical null hypthesis is H0: (mu)d = 0 where (mu)d is the mean difference between treatments. For this test, a difference is computed within every pair. The mean and standard deviation of these differences are computed and used in computing the test statistics.

Alternative hypothesis

The hypothesis that the researcher wants to prove or verify; a statement about the value of a parameter that is either "less than", "greater than", or "not equal to".

Margin of error for 95% confidence

The maximum amount that a statistic value will differ from the parameter value for the middle 95% of the distribution of all possible statisitcs. (Note: 95% can be changed to any other level of confidence.) This only accounts for sampling variability.

Is n the number of observations or the numbers of pairs in a matched pairs data sample?

The number of pairs

Level of Confidence can be defined as...

The percentage of the time that the procedure will produce intervals that contain the parameter value

Beta

The probability of failing to reject a false nell hypothesis (probability of a type II error)

p-value

The probability of getting data (summarized with the test statistic) as extreme or more extreme than th one observed (in the direction of the alternative hypothesis) assuming H0 is true.

Level of Significance (symbolized by alpha)

The probability of rejecting a true null hypothesis; equivalently, the largest risk a researcher is willing to take of reecting a true null hypothesis.

Alpha

The symbol for level of significance (probability of type I error)

What is the purpose of a confidence interval?

To provide plausible values that a parameter could be

T or F: For fixed α, increasing sample size increases power.

True

True or False: A 95% confidence interval can be used to perform a two-sided test on H0: μ = μ0 versus Ha: μ ≠ μ0 at α = 0.05.

True

True or False: A large difference between the observed statistic and the null value results in a small p-value.

True

True or False: A two-sample t procedure should NOT be used to analyze matched pairs data.

True

True or False: The margin of error, m, represents the maximum estimation error for a given level of confidence.

True

True or False: The null hypothesis is a statement of no difference.

True

True or False: We always assume H0 is true when we compute a p-value.

True

True or False: When computing a confidence interval, two conditions must be met: first, the sample must be random; second, the sample size must be greater than 30 or the sample must come from a normally distributed population.

True

True or False: When results are practically significant, they are also statistically significant.

True

Inference

Using results about saple statistics to draw conclusions about population parameters.

Type II error

When you accept a false H0 (null hypothesis)/fail to accept H0

Type I error

When you reject a true H0 (null hypothesis)

What is the quantity z∗?

a confidence multiplier

Standard error of p-hat

a measure of the variability (spread) of the sampling distribution of p-hat (STANDARD ERROR AND STANDARD DEVIATION ARE THE SAME THING)

Standard deviation of p-hat

a measure of the variability (spread) of the sampling distribution of p-hat.

41. Which one of the following is NOT part of the definition for P-value? a) Probability that the null hypothesis is true. b) Probability of obtaining a value of the statistic. c) The value of the statistic is farther from the claimed parameter value than the observed statistic. d) The null hypothesis is assumed to be true

a) Probability that the null hypothesis is true.

Sample results are said to be statistically significant whenever: a) the difference between the observed statistic and the claimed parameter value given in H0 is too large to be due to chance. b) the difference between the true situation and the observed situation could plausibly have resulted because H0 is false. c) the researcher subjectively classifies the observed deviation from what was expected under H0 as large enough to matter. d) the difference between the observed statistic and the claimed parameter value is large enough to be worth reporting

a) the difference between the observed statistic and the claimed parameter value given in H0 si too large to be due to chance

Fill in the Blank: The________hypothesis generally represents what the researcher wants to check, or suspects might actually be the case.

alternative

Confidence interval

an estimate of the value of a parameter in interval form with an associated level of confidence; in other words,a list of reasonable or plausible values for the parameter based on the value of a statistc. e.g. a confidence interval for my gives a list of possible values that my could be based on the sample mean.

Hypothesis testing (Test of Significance)

assessing evidence provided by the data in favor of or against some claim about the population.

The margin of error in a confidence interval covers only which kind of errors? a) interviewer errors b) errors due to random sampling c) bias errors due to wording of questions d) computational errors

b) errors due to random sampling

Which one of the following does NOT affect margin of error for a one-sample t confidence interval for μ? (Assume that the necessary conditions are met.) a) Level of confidence b) Sample size c) Standard error of x̄ d) Value of the parameter μ.

d) Value of the parameter μ.

Which of the following statements about the sampling distribution of the sample mean, x-bar, is not true? a. The distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is large enough. b. The distribution is normal regardless of the sample size, as long as the population distribution is normal. c. The distribution's mean is the same as the population mean. d. The distribution's standard deviation is smaller than the population standard deviation. e. All of the above statements are correct.

e. All of the above statements are correct

Point estimation

estimate an unknown parameter using a single statistic (e.g. x-bar, p-hat)

Interval estimation

estimate an unknown parameter using an interval of values that is likely to contain the true value of that parameter.

If the test statistic is negative, find area to the _______

left

compared to small samples, large samples have _____ variability

less

p

population proportion

P0

proportion under the null hypothesis

p*

proportion used in sample size calculations if you do not have a p-hat value, p*= .5

Statistically significant

results of a study that differ too much from what we expected to attribute to chance variation alone.

If the test statistic is positive, find area to the _______

right

P-hat

sample proportion

(mu)0 (mu null)

the claimed value of the population mean given in H0.

What is a test of significance intended to assess?

the evidence provided by data against the null hypothesis in favor of the alternative hypothesis.

t*

the multiplier of standarad error in computing margin of error for estimating a mean. The value for t* is found on the t table in the intersection of the appropriate df (degrees of freedom) row and level of confidence column.

Confidence level

the percent of the time thatthe confidence interval estimation procedure will give you intervals containing the value of the parameter being estimated. (Note: this can only be defined in terms of probablity as follows: --the probabiltty that the confidence interval to be computed (before data are gathered) will contain the value of the parameter. After data are collected, level of confidence is no longer a probability because a calculated confidence interval either contains the value of the parameter or it doesn't.)

We say a point estimator is unbiased if:

the sampling distribution is centered exactly at the parameter it estimates

What does the standard error of xbar estimate?

the standard deviation of the sampling distribution of x

T or F: The mean of every t-distribution is zero just like the standard Normal distribution.

true

T or F: The shape of the t-distribution gets closer and closer to the shape of standard Normal distribution as the degrees of freedom increase.

true