01:220:322 Chap.2 Probability and Statistics Review
Probability and Statistics Review
Why a statistics review comes first
Regression inference is ordinary statistical inference applied to a slope. The review lecture rebuilds the prerequisites from the textbook appendices so that tests and intervals on regression coefficients later feel familiar.
The notes even write the population mean as a beta, which is the same symbol later used for regression parameters.
Expected value, variance and covariance
The expected value is a probability-weighted average and is linear. The variance measures spread and scales with the square of any constant multiplier.
Covariance measures whether two variables move together; independence implies zero covariance but zero covariance does not imply independence.
The variance of a sum includes twice the covariance, a term students often drop without justification.
The sample mean as an estimator
The sample average is unbiased for the population mean, and under random sampling its variance is the population variance divided by the sample size. Dividing by n minus one makes the sample variance unbiased.
Consistency says the estimate gets close to the truth as the sample grows, and the central limit theorem says the standardised average is approximately standard normal in large samples, which is why the course uses normal critical values.
Tests, p-values and intervals
A test states a null and an alternative about the population parameter, computes a t statistic, compares it with a critical value from the chosen significance level, and decides.
One-sided and two-sided alternatives use different critical values. The p-value is the highest significance level at which the null survives.
A confidence interval collects the null values a two-sided test would not reject.
What gets examined
Exam papers repeatedly ask for a full test of a population mean from a sample mean, a sample variance and a sample size, including the decision at two different significance levels and the p-value.
Writing hypotheses correctly
Put the claim you want to overturn in the null, together with the equality.
If a transit agency claims commutes are at least thirty minutes and you suspect they are shorter, the null is that the mean is thirty or more and the alternative is that it is less. The tail of the rejection region follows the alternative, and a one-sided p-value is not doubled.
What this chapter covers
- 01
Expected value, variance and covariance
- 02
Variance of a sum
- 03
Unbiasedness and consistency
- 04
Central limit theorem
- 05
One-sided and two-sided tests
- 06
Confidence intervals and p-values
Worked example · free
A two-sided test of a mean
- 2Hypotheses: H0 says the mean equals 30; H1 says it differs from 30.
- 2Standard error = square root of 110.25/49 = 1.5, so t = (34.1 - 30)/1.5 = 2.73.
- 2Reject if the absolute t exceeds 1.96; 2.73 does, so reject at 5%. Two-sided p = 2P(Z > 2.73) = 0.006.
- 2Interval: 34.1 plus or minus 1.96 times 1.5 gives 31.16 to 37.04, which excludes 30.
Key terms
- Unbiased Estimator
- An estimator whose expected value equals the parameter it estimates.
- Consistency
- The property that an estimator gets arbitrarily close to the true value with high probability as the sample grows.
- Central Limit Theorem
- The result that a standardised sample average is approximately standard normal in large samples.
- Critical Value
- The cutoff a test statistic is compared with, set by the significance level and the alternative.
- P-Value
- The highest significance level at which the null hypothesis survives the test.
- Covariance
- A measure of whether two random variables tend to move in the same direction.
- Significance Level
- The probability of rejecting a true null hypothesis that the tester is willing to accept.
Probability and Statistics Review FAQ
When do I use 1.645 rather than 1.96?
Use 1.645 for a one-sided alternative at 5%, where all the rejection probability sits in one tail. A two-sided alternative splits the 5% across two tails, which pushes the cutoff out to 1.96.
Does a correlation between observations bias the sample mean?
No. Linearity of expectations keeps the average centred on the population mean whatever the covariances. Correlation changes the variance of the average, making it larger when the covariance is positive.
Why can a test reject at 5% but not at 1%?
The stricter level needs stronger evidence, so its critical value is further out. A statistic that lands between the two cutoffs is significant at the looser level only; the p-value sits between them.
Why do we divide the sample variance by n minus one?
Deviations are measured from the sample mean, which is itself estimated from the same data, so they are slightly too small on average. Dividing by n minus one corrects for that and makes the estimator unbiased.
Exam move
Drill the four-step test until it is automatic, then practise switching between one-sided and two-sided versions of the same data and explaining why the decision can change.
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