Stats 1 Midterm 2 Review

Hypothesis

prediction that is tested in a research study.

Probability Value

The probability of an outcome given the hypothesis. It is not the probability of the hypothesis given the outcome.

Null Hypothesis

• The hypothesis that an apparent effect is due to chance

• In other words, the idea that nothing is going on: there is no effect of our
  treatment, no relationship between our variables, and no difference in our sample mean from what we expected about the population mean.

• Written as H₀

• Either =, <, or >

• Typically the opposite of the researcher’s hypothesis (because you hope to reject it)

Why is the value mew (population parameter) used instead of M (sample statistic)? (Two reasons)

1. Once we collect data, we know what the value of M is—it’s not
   a mystery or a question, it is observed and used for the second reason

2. We are interested in understanding the population, not just our sample

Alternative Hypothesis

• The reverse of the null hypothesis; what you are trying to research

• Represented as H_A

• Stated as ≠, ≤, ≥

• It is easiest to start with stating before the null hypothesis

Significance Level (alpha)

• The probability value below which the null hypothesis is rejected is called the a level or simply a (“alpha”)

• Either .05 or .01; assume that alpha = .05 if not specified

Why do we set the significance level value? Why is this important?

We set this value beforehand to avoid biasing ourselves by viewing our results and then determining what criteria we should use. If our data produce values that meet or exceed this threshold, then we have sufficient evidence to reject the null hypothesis; if not, we fail to reject the null (we never “accept” the null).

Rejection Region/Critical Region

Our significance level corresponds to the area in the tail that is exactly equal to
alpha. If we use our normal criterion of a = .05, then 5% of the area under the curve becomes what we call the rejection region (also called the critical region) of the distribution.

Critical Value

The value corresponding to a specific rejection region (e.g. t_CRIT, z_CRIT)

How can you determine whether a test is one-tailed or two-tailed?

• If you are looking for whether there is a difference in something, but whether it is higher or lower is not specified, it is a two tailed test (e.g. a researcher wants to know if there is a difference in depression levels of individuals who go to therapy vs individuals who do not)

• If you are looking for whether one group of individuals are associated with higher or lower traits, it is one tailed (e.g. a researcher wants to know if individuals who seek therapy have lower levels of depression than those who do not attend therapy).

Characteristics of the z distribution

• Has a fixed, standard normal distribution

• Mean = 0

• SD of 1

• Has thinner tails than a t distribution

Test Statistic

Any z score falling outside ±1.96 (greater than 1.96 in absolute value) falls in the rejection region. When we use z scores in this way, the obtained value of z (sometimes called z obtained and abbreviated zobt) is something known as a test statistic, which is simply an inferential statistic used to test a null hypothesis

p value

The proportion of the area under the curve in the tails beyond our obtained z statistic.

What does it mean if the p value is less than alpha?

We reject the null hypothesis

What does it mean if the p value is greater than alpha?

We fail to reject the null hypothesis

Statistical Significance

When the null hypothesis is rejected, the effect is said to have statistical significance, or be statistically significant.

Steps in the Four-Step Procedure

1. State the hypotheses

2. Set decision criteria

3. Collect and compute

4. Conclusion

Effect Size

• Gives us an idea of how large, important, or meaningful a statistically significant effect is.

• Provide important information and clarification that overcomes
  some of the weakness of hypothesis testing

• Calculated using Cohen’s d

How do you interpret the Cohen’s d value?

0.20 = small

0.50 = moderate

0.80 = large

Type I Error

Occurs when a significance test results in the rejection of a true null hypothesis.

Type II Error

Failing to reject a false null hypothesis

Statistical Power

The probability of correctly rejecting a false null hypothesis equals 1 − β

1 - α

The probability of correctly failing to reject a true null hypothesis

t tests

• Used when you don’t know the population’s standard deviation (σ)

• We estimate the population’s standard deviation by using the sample statistic (s)

• Uses the statistic ‘t’ instead of ‘z’

• The process of calculating is exactly the same, but just with this replacement

How do we find the critical values (t)?

• We cannot use the z table, but we use the t table instead

• Find the df and the alpha level and look on the t table to find t_CRIT

Characteristics of the t distribution

• It is symmetric

• It is bell-shaped (not normally distributed)

• It becomes more leptokurtic as the sample size decreases

• Its shape depends on the degrees of freedom in the sample

• Small df = wider tails and more spread out

• Large df = narrower and more closely resembles z distribution

• Heavier tails

• Mean of 0, SD that is dependent on the df

How should you round if a problem requires you to find critical values and the exact degrees of freedom is not listed?

Round down to the smallest number because it avoids inflating Type I error

Point Estimate

a single value calculated from a sample that is used to estimate an unknown population parameter

Confidence Interval

Starts with our point estimate and then creates a range of scores considered plausible based on our standard deviation, our sample size, and the level of confidence with which we would like to estimate the parameter

Margin of Error

• The range of scores considered plausible based on our standard deviation, which extends equally in both directions away from the point estimate, is called the margin of error.

• It can only be calculated using the critical value for a two-tailed test since the margin of error moves away from the point estimate in BOTH directions — a one-tailed value does not make sense

How can we use confidence intervals to test our hypothesis?

• If the range of the confidence interval brackets (or contains, or is around) the null hypothesis value, we fail to reject the null hypothesis

• If it does not bracket the null hypothesis value (i.e., if the entire range is
  above the null hypothesis value or below it), we reject the null hypothesis.

Related Samples

• Used to research change over time

• Involve measuring a single variable at different times to see if we get the same score the second time as we did the first time

• Involves using the difference of the scores from Time 1 and Time 2 to compute values

Matched Pairs

we can also test to see if people who are matched or paired in some way agree on a specific
topic.