Intro Stats II Final

• 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 point estimate, and it is stated that this interval is likely to contain unknown population parameter(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%

In repeated sampling under identical conditions, (1-α)100% of all confidence intervals constructed in this manner will enclose the unknown mean μ

• 1. Confidence level (1-α)100% or z_α/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)

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.."

We can say with probability (1-α)100% that the maximum error is within ±E when estimating μ by x-bar

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 H_A , and defined as the complement or negation or opposite of the Null Hypothesis (H₀)

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.

denoted by β and represents to probability of accenting H₀ given H₀ is is false. Note: β≠(1-α)!

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.

• I. Formulate H₀ and H_A
• II. Select an appropriate test statistic (z_calculated )
• III. Fix the level of significance (α) and formulate the decision rule
• IV. Write your conclusion in words

Aka the Critical Region or Rejection Region, depends on H_A and α. If H_A is two-sided, we use z_α/2 and -z_α/2 or t(n-1, α/2) and -t(n-1, α/2). Otherwise we use z_α or -z_α in the same direction of H_A

The same units!

• An alternate method to test H₀ , the P-value is the probabillity, assuming H₀ is true that the statistic z_c 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.

• a) The Classical or Critical Value Approach
• b) the p-value Approach
• c) If H_A 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)

• 1) Two samples are random & independent
• 2) Both samples came from two independent, normal populations
• 3) σ₁² (σ₁) and σ₂²(σ₂) are known

• 1) Two samples are random & independent
• 2) Both samples came from normal populations
• 3) σ₁² (σ₁) and σ₂²(σ₂) are unknown but equal

s_p²

we combine the information given in both samples to compute estimated variance of p₁ & p₂

we do NOT combine the information contained in both samples to compute the estimated variance

Tests the Null Hypothesis that the observed frequencies follow a pattern or theoretical distribution. The test is goodness-of-fit because the hypothesis tested is how good the observed frequencies fit a given pattern

used to test whether of not the sampled multinomial data is in agreement with the hypothesized distribution. OR Testing 3 or more unknown population proportions.

A good agreement between the observed and expected frequencies results in a small value of . A perfect agreement would result in =0. Thus the Null Hypothesis is rejected if is large [upper tail test]

INDEPENDENT or not related (HoI)

DEPENDENT or related (HAD)

H₀: P₁=P₂ vs H_A:P₁≠P₂

• A test of homogeneity involves testing the
• H₀: the proportions of elements with certain characteristics in two or more different populations are the same
• vs
• H_A: the proportions are not the same

A procedure used to test the Null Hypothesis that the means of three of more populations are equal.

positive, since it's they're the sum of squares

Mean Square within (MSW)

• a. k-samples are random and independent
• b.Each of the k-samples came from a normal population
• c. σ²₁, σ²₂, σ²₃, ,σ²_k are unknown but equal

positive

positive or negative

, where

write the equation out in general form first. If there it is "a-bx" that means "a+(-b)x" which implies b is negative

denoted by and defined as the strength of the relationship between two variables, x&y.

denoted by r. Since the population correlation is usually unknown, the point estimator of the population correlation is the sample correlation r.

-1<r<1

we have a perfect relationship between x&y in a positive way. That is, if x increases, y increases.

if x increases, y decreases. Or if x decreases, y increases

the variables x&y are not related

denoted by for the population and r² for the sample. But or is usually unknown. The coefficient of determination for the samples represents the % of variation in the dependent variable y, explained by the independent variable (x) in the extimated least squares regression model ŷ. The higher the value of r² the better it is.

r≈1 for values of r>80% or .8