Randomization Slopes A randomization distribution is created...

Name: Problem 166, Chapter 11: Statistics Unlocking the Power of Data
Uploaded: 2022-02-28T23:38:35-08:00
Duration: 2 min 10 s
Channel: Sheryl Ezze

Randomization Slopes A randomization distribution is created to test a null hypothesis that the slope of a regression line is zero. The randomization distribution of sample slopes follows a normal distribution, centered at zero, with a standard deviation of 2.5 (a) Draw a rough sketch of this randomization distribution, including a scale for the horizontal axis. (b) Under this normal distribution, how likely are we to see a sample slope above $3.0 ?$ (c) Find the location of the $5 \%$ -tile in this normal distribution of sample slopes.

Randomization Slopes A randomization distribution is created to test a null hypothesis that the slope of a regression line is zero. The randomization distribution of sample slopes follows a normal distribution, centered at zero, with a standard deviation of 2.5
(a) Draw a rough sketch of this randomization distribution, including a scale for the horizontal axis.
(b) Under this normal distribution, how likely are we to see a sample slope above $3.0 ?$
(c) Find the location of the $5 \%$ -tile in this normal distribution of sample slopes.

Key Concepts

Percentiles and Tail Probabilities

Calculating percentiles and tail probabilities in a normal distribution involves determining the probability of observing a value above or below a certain cutoff. For example, finding the likelihood of a sample slope above a threshold or identifying specific percentiles, such as the 5% quantile, utilizes the properties of the normal distribution and z-scores.

Regression Slope

The regression slope represents the relationship between an independent variable and a dependent variable in a regression model. Testing whether the slope is zero examines whether there is a statistically significant linear relationship between the variables.

Randomization Distribution

A randomization distribution is created by repeatedly sampling under the null hypothesis to generate a distribution of a test statistic, such as a regression slope. It represents the expected variability in the statistic when the null hypothesis is true and is used to assess the significance of the observed value.

Null Hypothesis Testing

Null hypothesis testing involves stating a default assumption—in this case, that the regression slope is zero—and then determining whether the observed data provide sufficient evidence to reject this assumption. The procedure often uses simulated or theoretical distributions to calculate p-values.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, forming a bell-shaped curve. In this context, the randomization distribution of sample slopes is modeled as a normal distribution with a mean of zero and a specified standard deviation, allowing for the use of standard normal probability calculations.

Transcript

Please give Ace some feedback