BUAD 310 Exam 2

Point Estimate

• A single value to approximate an unknown population parameter

Sampling Error (or Sampling Variability)

• How much an estimate will tend to vary from one sample to the next

Elements for Constructing Confidence Intervals

1. Mean: center

2. Standard Error: spread

3. Critical Value (or z-score): scaling factor

What does 95% confidence interval mean?

• A 95% confidence intervals means that if we were to take many random samples from a population and calculate a confidence interval for each sample, approximately 95% of those intervals would contain the true population parameter (e.g. the true proportion)

Confidence Intervals

• Each possible sample gives us a different sample proportion and a different interval

• Even though the results vary from sample-to-sample, we are “confident” because the margin-of-error would be satisfied for 95% of all samples (with z = 2)

Standard Error

• The standard error is the square root part of the CI NOT including the multiplier (critical value) for your confidence interval

Margin for Error

• The margin for error is the multiplier (critical value) for your confidence intervals TIMES the square root part

• IT INCLUDES BOTH

Common Misinterpretations of Confidence Intervals

1. There’s a 95% probability that the true parameter lies in this confidence interval

2. 95% of confidence interval means 95% of the data fall inside the confidence interval

3. A wider confidence intervals means the estimate is more accurate

4. If the confidence interval contains 0, the true parameter is zero

5. The true parameter moves depending on the confidence interval we compute

Confidence Interval vs. Hypothesis Testing

• Confidence Interval: provide a range of values for the population parameter

  • (e.g. I’m confident that my cholesterol level is between 180 to 200 mg/dL)

• Hypothesis Testing: I have a single observation, how significant is it?

  • (e.g. I observe that my cholesterol level is 196 mg/dL, is it significantly higher than my average cholesterol level?

Use confidence interval when:

• Goal: estimate population parameter

• Describe uncertainty around the estimate

• Output: a range of values

Use hypothesis testing when:

• Goal: test a specific claim about a population parameter

• Compare groups or conditions

• Assess significance of results

• Output: a single value (p-value)

Hypothesis Testing Terminology

• Null Hypothesis: a statement that assumes no effect or difference exists in a population

• Alternative Hypothesis: a statement that assumes there is a statistically significant effect or difference in a population

  • DONT USE p HAT IN HYPOTHESIS STATEMENTS

  • Use the parameter that the study is interested in answering (usually population parameter)

P-Value

• If p-value > alpha, the p value is NOT statistically significant and we FAIL TO REJECT the null hypothesis

• If p-value < alpha, the p value is statistically significant and we REJECT the null hypothesis

• A p-value is the probability of an observed (or more extreme) result assuming that the null hypothesis is true

One-sided Hypothesis Test

• When the alternative hypothesis is > or < (only focusing on one tail of the distribution)

Two-sided Hypothesis Test

• When the alternative hypothesis is a does not equals (focusing on two tails of the distribution)

T-Test

• A t-test always centers at 0 and has a single parameter: degrees of freedom

• Degrees of Freedom: the precise form of the bell-shaped curve (n-1)

• YOU CAN TELL ITS A T-TEST WHEN THEY GIVE U THE STANDARD DEVIATION

Degrees of Freedom

• How many data points are free to vary after you’ve used some of them to estimate parameters like the sample mean

• Degrees of freedom help adjust for the uncertainty introduced by using a sample

Sample Mean

• There are two explanations why the sample mean is higher than the recommended 8 schools:

  • The true population mean is different

  • The true population mean is 8, and the difference between the true population mean and the sample mean is simply due to natural sampling variability

Common Misunderstanding in Hypothesis Testing

1. Rejecting null hypothesis means null hypothesis is 100% false

2. I have a large p-value, it means null hypothesis is true

Z Test

• Sample proportion

• P value > 0.05: fail to reject null

• P value < 0.05: reject null

  • Usually requires a large sample

T Test

• Sample mean

• Degrees of freedom: form of bell curve —> uncertainty

• Help adjust for the uncertainty introduced by using a sample

• Flexible, works with small samples