BNAD Exam 2

Suppose you are performing a hypothesis test on μ and the value of sigma is known. The the 5% significance level, the critical values for a two tailed test are:

-z.025 and z.025

In a binomial experiment there are 2 possible outcomes (k=2) and the probability of success (p) and the probability of failure (1-p) equals 1(P1+p2=1)

A multinomial experiment results in one of k possible outcomes (k≥3) and the probabilities added together equals 1 (p1+p2+...+Pk= 1)

With monster df

Always round down for df

Hypothesis Testing

Enables us to determine if the collected sample data is inconsistent with what is stated in the null hypothesis resolves conflicts between two competing opinions

Type II Error

Fail to reject the null hypothesis when the null hypothesis is false Denote the probability with β Miss

Null Hypothesis

Ho the presumed default state of nature or status quo equal sign goes in the null

The Goodness of Fit Test for Normality

Ho: Follows a normal distribution Ha: does not follow a normal distribution

What are the competing hypotheses if the thought that the proportion in group 1 watches TV more than in group 2?

Ho: p1-p2≤.20 HA: p1-p2>.20

Null Hypothesis for a Multinomial Experiment

Ho: p1=p2=p3=p4=0.25 Ha: not all population proportions are equal to .25 or Ho: p1=0.40, p2=0.30, p3=0.20, p4=0.10 HA: at least one of the proportions is different from its hypothesized value

You want to determine if proportion is more than .15

Ho: p≤ .15 HA: p>.15

The null hypothesis for a two sided test for a population mean would be noted as

Ho: μ=μ0

Hypothesis Tests for μD (Mean Difference)

Ho: μD=0 (there is no difference) HA: μD≠μ0 (there is a difference) reject the null if the range of the confidence interval does not include zero if the confidence interval includes zero we do not reject the null hypothesis and conclude that the mean does not differ from zero

Right Tailed Hypothesis Test

Ho: μ≤μ0 HA: μ>μ0 Alternative symbol points to the right

Left Tailed Hypothesis Test

Ho: μ≥μ0 HA: μ<μ0 Alternative symbol points to the left

Steps to Formulate the Null and Alternative

Identify the population parameter of interest Determine whether it is a one tailed or two tailed test Include some form of the equality sign in the null hypothesis and use the alternative hypothesis to establish a claim

Left Tail Probability of Alternative Hypothesis

P(Z≤z)

Right Tail Probability of Alternative Hypothesis

P(Z≥z)

Two Tail Probability of Alternative Hypothesis

P(Z≥z) if z>0 or P(Z≤z) if z<0

Confidence Intervals and Two-Tailed Hypothesis Tests

Reject H0 if the confidence interval does not contain the hypothesized value

Type I Error

Reject the null hypothesis when the null hypothesis is true denote the probability with α False alarm

P-Value

When testing μ, the probability of obtaining a sample mean at least as large or at least as small as the one derived from a given sample, assuming the null hypothesis is true

Multinomial Experiment

a generalization of a binomial experiment consists of a series of n independent and identical trials of a random experiment such that for each trial: -there are k possible outcomes or categories called cells; k≥2 -each time we repeat the trial the probability pi that one outcome falls into a particular cell remains the same -the sum of the cell cell probabilities is one, that is p1+p2+...+Pk= 1 Ho: p1+p2+...+Pk= 1

Matched-Pairs Sampling

a specific type of dependent sampling when the samples are paired in some way often a before and after study but does not necessarily need to be on the same individual parameter of interest is the mean difference we recognize by looking for a natural pairing between one observation in the first sample and one observation in the second sample

Test of Independence (Chi Square Test of a Contingency Table)

analyzes the relationship between two qualitative independent variables Ho: The two classifications are independent HA: The two classifications are dependent use a contingency table to conduct a hypothesis test that determines whether the classifications depend upon one another implemented as a right-tailed test and is valid when the expected frequencies in each cell are five or more

The basic principle for hypothesis testing is to first assume that the null hypothesis is true

and then determine if the sample data contradicts this assumption

α can only be reduced

at the expense of increasing β

We like to report the p-value

because people can make their own decisions about what α they want and whether or not they want to reject it

Binomial Distribution

can be approximated by a normal distribution for large sample sizes

Final conclusion to a statistical test

clearly interpret the results in terms of the initial claim

Two Tailed Hypothesis Test

defined for when the null hypothesis states a specific value for the population parameter of interest Ho: μ=μ0 HA: μ≠μ0 we can reject the null on either side of the hypothesized value of the population parameter

Goodness of Fit Test for a Multinomial Experiment

determines whether two or more population proportions equal each other or any predetermined set of values For example, are four candidates in an election equally favored by voters? or Do people rate food quality in a restaurant comparably to last year?

Degrees of Freedom for a Contingency Table

df=(r-1)(c-1)

Goodness-of-fit test

examines a single qualitative variable the chi-square test statistic will be at least zero

A required condition of Chi Test

expected frequency in each cell must be at least 5 One way to correct this potential problem is to combine categories

We can only do confidence intervals

for two tailed hypothesis tests

Both α (Type I Error) and β (Type II Error) will decrease

if n increases it is always in the best interest to have a sample size as large as you can possibly afford

two (or more) random samples are considered independent

if the process that generates one sample is completely separate from the process that generates the other sample

In the case when we construct a confidence interval for μ1-μ2 where σ²1 and σ²2 are unknown but assumed equal we calculate a pooled estimate of the common variance s²p

in the case where the variances are not assumed equal, we cannot calculate a pooled estimate of the population variance because of different variabilities in the two populations

One Tailed Hypothesis Test

involves a hypothesis that can only be rejected on one side of the hypothesized value (you're guessing that the mean is greater than (left tailed) or less than (right tailed) a certain value) α=.05 Z=1.645 α=.01 Z=2.33

d0

is a hypothesized mean difference if an interval does not include d0 we reject the null

The point estimate for the difference between two population means

is represented by the difference between two sample means

In the case when we construct a confidence interval for the difference between two proportions p1-p2 follows the general format of a point estimate ±

margin of error

Null and Alternative

mutually exclusive

Hypothesis testing for the difference between two sample means is only valid when the sapling distribution of X1-X2 is

normally distributed

The normal distribution approximation for a binomial distribution is valid when

np ≥ 5 and n(1-p) ≥ 5

The p-value is calculated assuming the

null hypothesis is true

When performing a hypothesis test on μ, the p-value is defined as the

observed probability of making a Type I error

Equivalent methods to test a one sided hypothesis

p-value approach critical value approach the conclusions are always the same; only the decision rules defer between the two approaches

When comparing two population proportions the parameter of interest is

p1-p2

A pooled estimate of proportions can be used when the underlying sample proportions are essentially the same estimates of the unknown

population proportion

Alternative Hypothesis

represented with HA the results that you want a contradiction of the default state of nature or status quo

The optimal values of Type I and Type II errors

require a compromise in balancing the costs of each type of error

The hypothesized value of the mean

resides in the null

Independent Random Samples

samples that are completely unrelated to one another; the process that generates one sample is completely independent of the process that generates the other sample

Contingency Table

shows the frequencies for two qualitative variables, x and y, where each cell of the table represents a mutually exclusive combination of the pair of x and y values

The power of a test

the probability of rejecting the null hypothesis when the null hypothesis is false the probability of being correct 1-β or 1-(the probability of not rejecting the null when it is false)

Statistical inference concerning the mean difference based on matched-pairs sampling requires one of two conditions

the sample size n≥30 The paired difference D=X1-X2 is normally distributed

In a test of independence

the test statistic follows the X^2df distribution

Given a right tailed hypothesis test, if the value of the test statistic is 1.82 and the critical value is 1.645

then we reject the null hypothesis

Inferential Statistics

use sample information to make decisions about an unknown population parameter

We calculate pooled estimate of the common variance by

using weighted averages of the sample variances

In order to conduct a test of independence

we calculate each cell's probability in a contingency table by applying the multiplication rule for independent events

When p<α

we reject the null

If a test statistic falls into the rejection region

we reject the null hypothesis

For most applications the hypothesized difference between two means is

zero

Significance Level

α the allowed probability of making a type I error choose the value before conducting a test

The parameter of interest for matched pairs sampling

μD

Mean Difference for Matched Pairs Sampling

μD= X1-X2 X1 and X2 are matched in a pair

We do not reject the null when the p value is

≥α