ST 311 Midterm 2 NCSU

Test Statistic

Test Statistic is a numeric measure of the distance from the sample value to what is expected under the Null hypothesis. It is based on a standard score.

Null Distribution

The Null Distribution is the sampling distribution of the test statistic assuming the null hypothesis is true.
The Null Distribution tells us how a statistic will act, and what is likely if the Null hypothesis
is true (p-value).

P- Value

Proportion of the null distribution that is more
extreme than the observed test statistic.
P-Value tells us how likely our statistic is, if the Null hypothesis is true. If the p-value is small then the statistic is unlikely given the null hypothesis. Found in Table Z in the direction of the alternative hypothesis (Ha).

Alternative Hypothesis (Ha): parameter < null value

Ha: parameter < null value: area below test statistic

Alternative Hypothesis (Ha): parameter > null value: area above test statistic

Ha: parameter > null value: area above test statistic

2 tailed

Ha: parameter not equal sign null value: area in both tails

Significance Level

The Significance Level is a cut off point for determining if a p-value is "small". Called α or "alpha." Common values are 0.01, 0.05, and 0.1

If the p-value is less than α, then..

If the p-value is less than α, then reject the Null hypothesis.

If the p-value is greater than α, then..

If the p-value is greater than α, then do not reject the Null hypothesis. Never accept the Null Hypothesis.

4 Basic Steps of Hypothesis Testing (ECTNPI)

1. Establish null and alternative hypotheses.
2. Check necessary conditions.
3. Calculate test statistic, specify null distribution and find p-value.
4. Interpret p-value and make conclusion

When conducting a hypothesis test, everything is
done assuming that...

When conducting a hypothesis test, everything is done assuming that the null hypothesis is true.

If we have a large random sample, the sample proportion or p-hat will be...

If we have a large random sample, the sample proportion (p-hat) will be Normally Distributed,
Centered at the True Population Proportion (p)
and have a standard deviation of the square root of p times q divided by n.

The Standard Normal Distribution (Z-dist) is..

The Standard Normal Distribution (Z-dist) is the Null Distribution.

Matched Pairs

Each unit is measured twice. The units can be:
A single subject (serves as its own control) examples include: Before & after, Right & left hand.
OR
Similar units are matched/paired together prior to a study, pair subjects with similar characteristics through natural connection (sex, age, health, etc.)

When using inference with paired data, the population parameter of interest is...

When using inference with paired data, the population parameter of interest is the true mean of the differences.

Statistical Significance

statistically significance means it is unlikely that the results were simply due to random chance.

Practical Significance

Practical significance is about whether or not the results actually make a difference in a practical sense or a real-world setting. Subjective, not always clear answer.

Confidence Interval

A confidence interval contains the range of legitimate values for the parameter. You can use a confidence interval (rather than a test-statistic and p-value) to test a two sided alternative hypothesis about a parameter.

If the null value is contained IN the confidence interval then...

If the null value is contained IN the confidence interval then: do NOT reject the Null Hypothesis (Ho).

If the null value is NOT IN the confidence interval then...

If the null value is NOT IN the confidence interval then: do reject the Null Hypothesis (Ho).

A 95% confidence interval is equivalent to testing using a BLANK level of significance.

A 95% confidence interval is equivalent to testing using a 5%
level of significance.

A 99% confidence interval is equivalent to testing using a BLANK level of significance.

A 99% confidence interval is equivalent to testing using a 1%
level of significance.

Regression

Regression summarizes the relationship between two variables. Used to estimate the average of one variable for a particular value of the other variable, while noting that individual values
will vary around this average.

Response Variable (DV) (y)

Response variable is what we wish to learn about. The Response Variable is called the outcome or Dependent Variable.
Notation = y

Explanatory Variable (IV) (x)

Explanatory Variable explains the
changes we see in the Response Variable.
The Explanatory Variable is called the Independent variable.
Notation: x

Scatter Plot

A Scatter Plot is a graph used to show the relationship between the Response variable (y, plotted on the vertical axis) and Explanatory variable (x, plotted on the horizontal axis).

Examining A Scatter Plot

1. Overall pattern or form of the relationship:
is it a straight line (linear) or not?

2. Direction of relationship: Positive or negative?
Positive: Variables increase or decrease together
Negative: As one variable increase, the other decreases.

3. Strength of relationship: How much do points vary around the overall pattern? Strong (points close together) or Weak (points scattered)?

4. Deviations from overall pattern: any outliers?

Correlation Coefficient (r)

Measures only the strength of the Linear
relationship between y and x. Ranges between +1 and -1. Value doesn't depend on which variable is the Response Variable or which is the Explanatory Variable (If I switch x and y, I should still get the same r value). Does not depend on units of measurement (in, cm). IS sensitive to outliers.
Notation: r

Regression Line aka Least Squares Line

yˆ = b0 + b1x

yˆ = Predicted (estimate or fitted) value of y.
x = Explanatory variable.
b0 = Intercept (average value of y when x = 0).
b1 = Slope (change in average value of y for a 1-unit change in x).

Residual (observed - expected)

ei = yi − yˆi

Vertical distance between the regression line and a particular data point. Note that this value could be positive or negative, depending
on if it falls above the line or below it.

R2 (r squared) aka Coefficient of Determination

(the square of the correlation r): Measures the proportion of the variation in y that can be explained by its linear relationship with x. Ranges from 0 to 1.