Statistical Inference Slide 3: The expectation (mean) of the estimator β1 hat is equal to the true parameter β1, and the variance of β1 hat is σ^2 / S_XX. This is derived by expressing β1 hat as a linear combination of the response variables (Yi) and using the properties of expectations. Unbiasedness refers to the fact that the expectation of the estimator is equal to the true parameter being estimated. Slide 4: Under the assumption of normality, β1 hat follows a normal distribution. The expectation and variance of β1 hat are established, providing the statistical properties of the estimator. Slide 5: The sampling distribution of β1 hat is not directly known because the variance of the error terms (σ^2) is unknown. To obtain a complete specification of the distribution, the estimator σ^2 is used. This estimator follows a chi-square distribution with (N - 2) degrees of freedom. As a result, the sampling distribution of β1 hat becomes a t distribution with (N - 2) degrees of freedom. Slide 6: The confidence interval for β1 is derived based on the assumption of normality. The confidence interval is centered around the estimated value of β1 and determined by the critical point from the t distribution. The critical point is influenced by the desired confidence level and accounts for the variability of the estimator. Slide 7: The confidence interval formula is further explained, indicating that the sampling distribution of β1 hat is a t distribution. To construct a confidence interval with (1-alpha)% confidence level, the critical point from the t distribution is used. The critical point incorporates the confidence level and the standard deviation of β1 hat. Slide 8: Hypothesis testing for β1 involves comparing the estimated value of β1 to a specified null hypothesis, such as β1 = 0. The t-value is calculated as the difference between the estimated value and the null value divided by the standard error of the estimator. If the t-value is large, the null hypothesis is rejected, indicating that β1 is statistically significant. Slide 9: The procedure for testing whether β1 is equal to a constant (C) is explained. The t-value is computed similarly, but the null value is replaced with the constant of interest. If the t-value exceeds its critical point or the p-value is below a specified significance level (e.g., 0.01), the null hypothesis is rejected, indicating a statistically significant difference from the constant. Slide 10: If the interest is in testing whether the regression coefficient is positive or negative, the alternative hypothesis is changed accordingly. The p-value is then calculated based on the tail of the distribution corresponding to the specific alternative hypothesis. Slide 11: The inference for the intercept parameter (β0) is similar to that of the slope parameter (β1). The expectation of the estimator β0 hat is equal to the true parameter β0, making it an
unbiased estimator. The variance of β0 hat is determined, and the sampling distribution follows a t distribution. The confidence interval for β0 is similar to that of β1, centered around the estimator and using the critical point from the t distribution.
Statistical Inference Data Examples Slide 3: In this lesson, we focus on statistical inference for the regression coefficients in simple linear regression. We aim to make inferences on the estimated coefficients, including their statistical significance and the construction of confidence intervals. Slide 4: In the given example of the relationship between advertising expenditure and sales, the following inferences can be made: ●Estimated coefficient β1 and its variance, along with the sampling distribution of β1. ●Estimated coefficient β0 and its variance, along with the sampling distribution. ●Statistical significance of β1, whether it is significantly different from zero. ●Whether β1 is statistically positive. ●Derivation of a 99% confidence interval for β1. ●Interpretation of the p-value in the context of hypothesis testing. Slide 5: The regression model output provides the estimated values of the coefficients, as well as their standard errors and p-values. The estimated value for β1 is given, and its estimated variance can be obtained by squaring the standard error. The sampling distribution for β1 is a t-distribution with the specified degrees of freedom. The estimated value and variance of the intercept coefficient are also provided, with the variance obtained by squaring the standard error. Slide 6: To test whether β1 is equal to zero, the p-value provided by R can be used. If the p-value is approximately zero, the null hypothesis of β1 being equal to zero is rejected, indicating that β1 is statistically significant. To test whether β1 is statistically positive, the alternative hypothesis is changed to β1 being greater than zero. The p-value can be computed using the cumulative distribution function (CDF) of the t-distribution. A small p-value supports the conclusion that β1 is statistically positive. The 'confint()' function can be used to estimate a confidence interval, providing intervals for both the intercept and slope coefficients. The interpretation of a confidence interval is that it captures the true parameter value with a certain level of confidence, and in this case, it is a 99% confidence interval. The p-value is a measure of the evidence against the null hypothesis, indicating the degree to which the null hypothesis can be rejected based on the observed data.