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Two parts of The Statistical Inferences
- 1. Estimating unknown parameter(s) and constructing (1-α)100% Confidence interval for unknown parameters
- 2. Tests of Hypothesis about the unknown parameter(s)
A rule that tells us how to calculate the estimator based on the information contained in the sample. It is generally expressed as a formula which does not involve any unknown parameters in it. There are two types of estimators: Point Estimator and Interval Estimator
An estimator given as a point or a single value
- Let theta-hat be the point estimator of unknown population parameter theta [where theta could be μ or p or σ2 ] if E(theta-hat)= theta, then the point estimator theta-hat is an unbiased estimator of theta
- eg. E(s)=μ
When an interval is constructed around the poitn estimate, and it is stated that this interval is likely to contain unknonw population paramete(s) with a specific level. This confidence level is usually denoted by (1-α)100% where α is called the coefficient of confidence. If (1-α)100% is not given, we usually use (1-α)100%=95%
Interpretation of (1-α)100% Confidence Interval
In repeated sampling under identical conditions, (1-α)100% of all confidence intervals constructed in this manner will enclose the unknown mean μ
3 quantities to decrease the width of the Confidence Interval
- 1. Confidence level (1-α)100% or zα/2 (not ideal, lowers the probability that our confidence interval contains the unknown mean μ)
- 2. Population variance σ2 or σ (not ideal, as recalculating variance is time consuming and costly, as we have to go through the entire population)
- 3. Sample size n (ideal)
Margin of Error (for the estimate of unknown mean μ)
Denoted by E and defined as the quantity that is subtracted and added to the sample mean to obtain (1-α)100% confidence interval. Also called the "bound on the error of estimation" or "the maximum error" or "the estimation is within.."
Interpretation of E
We can say with probability (1-α)100% that the maximum error is within ±E when estimating μ by x-bar
The most conservative estimate of n
When we have no prior information about p or q, we use p=.5 and therefore q=.5 so that the variance of p-hat, v(p-hat) is maximized.
The sample size n obtained using p=.5 and q=.5 is called the most conservative estimate of n.
A conjecture about the unknown population parameter(s). The conjecture may or may not be true. There are two types of statistical hypothesis for each situation, called the Null Hypothesis and the Alternative Hypothesis
Denoted by H0, and states that the unknown population parameter is equal to a specific value. The Null Hypothesis always has an equal sign in it, and this is the hypothesis that is actually tested.
Denoted by HA , and defined as the complement or negation or opposite of the Null Hypothesis (H0)
Type I error
denoted by α and represents the probability of rejecting H0 given H0 is true. The value of alpha is also called the significance level of the test.
Type II error
denoted by β and represents to probability of accenting H0 given H0 is is false. Note: β≠(1-α)!
The Power of the Test
1-β, where β is Type II error. Both β and α cannot be reduced simultaneously for fixed sample size n (one goes up when the other goes down). Increasing n maximizes the power of the test, as it lowers both β and α.
The Classical or Critical Value Approach to testing Hypothesis
- I. Formulate H0 and HA
- II. Select an appropriate test statistic (zcalculated )
- III. Fix the level of significance (α) and formulate the decision rule
- IV. Write your conclusion in words
The Decision Rule
Aka the Critical Region or Rejection Region, depends on HA and α. If HA is two-sided, we use zα/2 and -zα/2 or t(n-1, α/2) and -t(n-1, α/2). Otherwise we use zα or -zα in the same direction of HA
The mean and standard deviation always have...
The same units!
- An alternate method to test H0 , the P-value is the probabillity, assuming H0 is true that the statistic zc would take an extreme or mre extreme value than the actually observed value.
- In fact, the p-value is the smallest calculated α or Type I error assuming H0 is true. Thus, we reject H0 if α>p-value.
Three methods for Testing Hypothesis
- a) The Classical or Critical Value Approach
- b) the p-value Approach
- c) If HA is two-sided, (1-α)100% confidence interval for μ (ie, Reject H0 if μ=μ0 does not lie in the (1-α)100% confidence interval)