Statistics-Chapter 8

Confidence limits

The values that define the boundaries of the interval of the interval

Directional test

*Is designed to detect differences from a hypothesized population mean in on direction only. *Often referred to as one-tailed tests *In general, a directional test will be more powerful than a corresponding nondirectional test if the actual population mean and they hypothesized population mean differ in the specified direction.

T-score

*Analogous to a z score except that it represents the number of estimated standard errors that a sample mean is from the hypothesized value of mue.

T distribution

*Can be used to determine the probability that a result would occur by chance, given a true null hypothesis. *There is a family of t distributions *Unlike the normal distribution, the exact shape of the t distribution is influenced by the number of degrees of freedom that are associated with it, which is related to the number of scores in a sample. *All t distributions are similar to the normal distribution in that they are bell shaped and symmetrical *The mean of a t distribution is always 0.

Type II error

*Fail to reject the null hypothesis when it is false *Beta

Failing to reject the null hypothesis

*In principle, we can never accept the null hypothesis as being true based on statistical tests; we can only reject it as being untenable. *When the observed value of z falls within the range defined by the critical values, we fail to reject the null hypothesis.

One sample t test

*Is the same as the formula for calculating a z score but with s hat sub x bar (the estimated standard error of the mean) substituted for the actual standard error of the mean in the denominator.

Nondirectional test

*One that is designed to detect differences either above or below the hypothesized population mean by considering both alternatives to the null hypothesis (the population mean being less than the value specified in the null hypothesis and the population mean being greater than the value specified in the null hypothesis). *Referred to as two tailed tests *If the population mean differs from the hypothesized population mean in the opposite direction from that stated in the alternative hypothesis, a nondirectional test will be more powerful than its directional counterpart. *Researchers are often interested in detecting deviations from the null hypothesis regardless of their nature, so they use this type of test.

Type I error

*Reject the null hypothesis when it is true *Alpha

Alpha level

*Rejection regions are determined with reference to a probability value

Null Hypothesis

*The hypothesis of "no difference" *The hypothesis that we assume to be true for the purpose of conducting a statistical test.

Confidence Interval

*The interval to be constructed. *The construction of the confidence intervals differs somewhat depending on whether the standard error of the mean is known or has to be estimated from sample data.

Assumptions of the one-sample t test

*The one sample t test is appropriate when the variable being studied is quantitative in nature and measured on a level that at least approximates interval characteristics. 1) The sample is independently and randomly selected from the population of interest. 2) The scores on the variable are normally distributed in the population.

Statistical power

*The probability of correctly rejecting the null hypothesis when it is false

Rejection Region

*The set of all Z-scores more extreme than the critical values (that is, less then the negative critical value or greater than the positive critical value) *Constitutes an unexpected result.

Critical values

*z-scores that define the endpoints of a range.

Hypothesis Testing (general)

1) Investigator begins by stating a proposal, or hypothesis, that is assumed to be true. 2) Based on this assumption, an expected result is specified. 3) The data are collected, and the observed result is compared with the expected result. 4) If the observed result is so discrepant from the expected result that it is very unlikely that the difference is due to chance, then the original hypothesis is rejected. 5) Otherwise it is not rejected.

Competing Hypotheses

1) Null hypothesis 2) Alternative Hypothesis

Formal Hypothesis Testing Steps

1) Translate the research question into two competing hypotheses: A null hypothesis and an alternative hypothesis. 2) Assuming that the null hypothesis is true, state an expected result in the form of a range of values within which the sample mean would be expected to fall. -This is expressed in terms of the Z-scores (critical values and rejection region) 3) Characterize the mean and the standard deviation of the sampling distribution of the mean, assuming that the null hypothesis is true. 4) Convert the observed sample mean to a Z-score to determine how many standard errors it is from mue, assuming that the null hypothesis is true (this is accomplished using the formula for the one-sample z test. 5a) Compare the value of z that was calculated using equation 8.1 with the expected result as defined by all z scores that fall between the critical values of z. -If the observed z score exceeds either the positive or the negative critical value, reject the null hypothesis, if the observed z score does not exceed either the positive or negative critical value, do not reject the null hypothesis. 5b) If the null hypothesis is rejected, compare the observed sample mean with the value of mue stated in the null hypothesis.

Normality assumption

The scores on the variable are normally distributed in the population.

Under some conditions the one sample t test is robust to violations of the normality assumption;

When we say that a test is robust to violations of a distributional assumption, we mean that the frequencies of type I and type II errors and, thus, the accuracy of our conclusions, are relative ineffective compared when the assumption is met.