Stats Exam 3 Terms

Finite Population Correction Factor

-In some sampling situations, the sample size n may represent 5% or perhaps 10% of the total number N of sampling units in the population. When the sample size is large relative to the number of measurements in the population (see the next slide), the standard errors of the estimators of µ and p should be multiplied by a finite population correction factor. -Use the finite population correction factor when n/N > .05.

observed significance value (p-value)

-The observed significance level, or p-value, for a specific statistical test is the probability (assuming H0 is true) of observing a value of the test statistic that is as large as it is due to chance alone. -Smallest value of a for which H0 can be rejected -Used to make rejection decision: If p-value > or equal to a, do not reject H0 If p-value < a, reject H0

difference between the sampling distributions of t and z

-The t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0. -The primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic.

interval estimator (or confidence interval)

-a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter -Gives information about closeness to unknown population parameter -ex:Unknown population mean lies between 50 and 70 with 95% confidence

null hypothesis

-denoted H0, usually represents the "status quo" (value of a parameter of interest) against which some claim about the population parameter will be tested. -In essence, it holds that there is no difference between the "status quo" and the claimed value.

Conditions Required for a Valid Large-Sample Hypothesis Test for p

1. A random sample is selected from a binomial population. 2. The sample size n is large. (This condition will be satisfied if both np0 ≥ 15 and nq0 ≥ 15.)

Conditions Required for a Valid Small-Sample Hypothesis Test for µ

1. A random sample is selected from the target population. 2. The population from which the sample is selected has a distribution that is approximately normal.

Conditions Required for a Valid Small-Sample Confidence Interval for µ

1. A random sample is selected from the target population. 2. The population has a relative frequency distribution that is approximately normal.

Conditions Required for a Valid Confidence Interval for σ^2

1. A random sample is selected from the target population. 2. The population of interest has a relative frequency distribution that is approximately normal.

Conditions Required for a Valid Large-Sample Hypothesis Test for µ

1. A random sample is selected from the target population. 2. The sample size n is large (i.e., n ≥ 30). (Due to the Central Limit Theorem, this condition guarantees that the test statistic will be approximately normal regardless of the shape of the underlying probability distribution of the population.)

Conditions Required for a Valid Large-Sample Confidence Interval for µ

1. A random sample is selected from the target population. 2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of SD.

Conditions Required for a Valid Large-Sample Confidence Interval for p

1. A random sample is selected from the target population. 2. The sample size n is large. This condition will be satisfied if both np > or = 15 or nq > or = 15 . Note that np and nq are simply the number of successes and number of failures, respectively, in the sample.).

one tailed test

A one-tailed test of hypothesis is one in which the alternative hypothesis is directional and includes the symbol " < " or " >."

point estimator

A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter -provides a single value -gives no information about how close the value is to the unknown population parameter

two tailed test

A two-tailed test of hypothesis is one in which the alternative hypothesis does not specify departure from H0 in a particular direction and is written with the symbol " ≠."

confidence coefficient

the probability that a randomly selected confidence interval encloses the population parameter - that is, the relative frequency with which similarly constructed intervals enclose the population parameter when the estimator is used repeatedly a very large number of times

Adjusted (1 - a)100% Confidence Interval for a Population Proportion, p

where p^~ is the adjusted sample proportion of observations with the characteristic of interest, x is the number of units in the sample that have the characteristic, and n is the sample size. -We use this when p is near 0, or is near 1.

characteristics of alternative hypothesis

Opposite of null hypothesis The hypothesis that will be accepted only if the data provide convincing evidence to reject the null hypothesis. Designated Ha Stated in one of the following forms: 1) Ha: mean does not equal (some value) 2) Ha: mean is less than (some value) 3) Ha: mean is greater than (some value)

Power of Test

Probability of rejecting false H0: -Correct decision -Equal to 1 - B Affected by: -True value of population parameter -Significance level a -Standard deviation & sample size n

degrees of freedom

The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n - 1) degrees of freedom (df).

confidence level

is the confidence coefficient expressed as a percentage

Conditions Required for a Valid F-Test for Equal Variances

1. Both sampled populations are normally distributed. 2. The samples are random and independent.

Properties of B and Power

1. For fixed n and a, the value of B decreases, and the power increases as the distance between the specified null value µ0 and the specified alternative value µa increases. 2.For fixed n and values of µ0 and µa, the value of B increases, and the power decreases as the value of a is decreased. 3. For fixed a and values of µ0 and µa, the value of B decreases, and the power increases as the sample size n is increased.

Properties of Sampling Distribution of p

1. The mean of the sampling distribution of ^p is p; that is, ^p is an unbiased estimator of p 2. The SD of the sampling distribution ^p is sqrt(pq/n) 3. For large samples, the sampling distribution of ^p is approx. normal. A sample size is considered large if both n^p greater than or equal to 15 and n^q is greater than or equal to 15.

Conditions Required for Valid Inferences about (μ1 - μ2), Variance 1 and Variance 2 unknown

1. The two samples are randomly selected in an independent manner from the two target populations. 2. Both sampled populations have distributions that are approximately equal. 3. The populations variances are equal

Conditions Required for Valid Large-Sample Inferences about(p1 - p2)

1. The two samples are randomly selected in an independent manner from the two target populations. 2. The sample sizes, n1 and n2, are both large so that the sampling distribution of p1-p2 will be approximately normal. (This condition will be satisfied if both n1p1 > or equal to 15, n1q1 > or equal to 15 , and n2p2 or equal to 15, n2q1 > or equal to 15

Conditions Required for Valid Small-Sample Inferences about µd

1.A random sample of differences is selected from the target population of differences. 2.The population of differences has a distribution that is approximately normal.

Conditions Required for Valid Large-Sample Inferences about µd

1.A random sample of differences is selected from the target population of differences. 2.The sample size nd is large (i.e., nd ≥ 30); due to the Central Limit Theorem, this condition guarantees that the test statistic will be approximately normal regardless of the shape of the underlying probability distribution of the population.

Possible Conclusions for a Test of Hypothesis

1.If the calculated test statistic falls in the rejection region, reject H0 and conclude that the alternative hypothesis Ha is true. State that you are rejecting H0 at the a level of significance. Remember that the confidence is in the testing process, not the particular result of a single test. 2. 2. If the test statistic does not fall in the rejection region, conclude that the sample does not provide sufficient evidence to reject H0 at the level of significance.

Conditions Required for Valid Large-Sample Inferences about(μ1 - μ2)

1.The two samples are randomly selected in an independent manner from the two target populations. 2.The sample sizes, n1 and n2, are both large (i.e., n1 ≥ 30 and n2 ≥ 30). [Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of (x1- x2) will be approximately normal regardless of the shapes of the underlying probability distributions of the populations. Also, (s1)^2 and (s2)^2 will provide good approximations to and when the samples are both large.]

Type 1 Error

A Type I error occurs if the researcher rejects the null hypothesis in favor of the alternative hypothesis when, in fact, H0 is true. The probability of committing a Type I error is denoted by a.

Type 2 Error

A Type II error occurs if the researcher fails to reject the null hypothesis when, in fact, H0 is false. -The Type II error probability B is calculated assuming that the null hypothesis is false because it is defined as the probability of not rejecting H0 when it is false The probability of committing a Type II error is denoted by B. -The probability of committing Type II error is not easily estimated.

sampling error

In general, we express the precision associated with a confidence interval for the population mean µ by specifying the sampling error within which we want to estimate µ with 100(1 - a )% confidence. The sampling error (denoted SE), then, is equal to the half-width of the confidence interval

rejection region

The rejection region of a statistical test is the set of possible values of the test statistic for which the researcher will reject H0 in favor of Ha.

test statistic

The test statistic is a sample statistic, computed from information provided in the sample, that the researcher uses to decide between the null and alternative hypotheses.

level of significance

The value of a is usually chosen to be small (e.g., .01, .05, or .10) and is referred to as the level of significance of the test.

target parameter

Using sample information to estimate the population parameter of interest and to assess the reliability of the estimate. (ex:: mean or proportion)

alternative (research) hypothesis

represents the hypothesis that will be accepted only if the data provide convincing evidence to reject the null hypothesis. This usually represents the values of a population parameter for that is of interest to the researcher. Denoted Ha.

statistical hypothesis

statement about the numerical value of a population parameter