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Unformatted text preview: Final Exam: Potential Topics. Topics.
Estimation of Population Parameters: Sample mean, sample standard deviation (Ch. 3) Sample proportion (Ch. 11) Sampling Distributions for Sample Mean (Ch 7) Inference for Population Mean Confidence Intervals (Ch. 8) Hypothesis Tests (Ch. 9) 9) P-Values (Ch. 9) Two Sample Tests for Means (Ch. 10, Section 3) Inference for Population Proportions Confidence Intervals (Ch. 11) Hypothesis Tests (Ch. 11) Estimation:
Draw Draw a sample – a subset of the population. Estimate the Population Parameter: Estimate
Pop. Parameter Estimator
sX = μ σ π = ∑x
p= 2 − (∑ x) 2 n n −1 X n Sampling Distributions
Sampling Distribution for Sample Mean. Shape – Normal if: X is normal ; is normal n ≥ 30 (CLT); X symmetric and n fairly large (n > 15). Center: Mean of all possible sample means. Sampling Distributions
Sampling Distribution for Sample Proportion. Shape – Normal when: nπ ≥ 5 and n(1-π) ≥ 5 n(1Center: Mean of all possible sample proportions μp = π
Variation: Summarize sampling error for sample proportions Standard Error: σ p μX = μX Variation: Variability of sample means around the Var center – summarize sampling error. Standard Error: = π (1 − π )
n σX = σX
n 1 Sampling Distributions
Areas Areas under the sampling distribution. Use the zsampling ztable given a z-score for the sample mean: z- Inference: Inference: Point and Interval Estimation. Point Point estimation – a single numeric value from a sample using our estimator sample using our estimator. Estimators Estimators are random variables – their distributions are called sampling distributions. Confidence Confidence Intervals: Use point estimate and point information about the sampling distribution information about the sampling distribution. Confidence Confidence Intervals are also random. zx = X − μx σx = σX X − μx n Given zGiven a z-score or probability, solve for the sample mean: X − value : X = μ X + zα ⋅ σ x Interval Estimation. Choose Choose Level of Confidence: (1 - α) Prior to estimation, we have a (1 - α)·100% to estimation we have (1 chance of drawing a sample that will give us an interval that contains the true population parameter value. Confidence Intervals are random. Recall our exercises in class. Confidence Intervals:
Identify the Unknown Population Parameter: μx
σx known z-interval σx not known t-interval π
Always z-interval x ± zα / 2 ⋅ σ x x ± tα / 2 ⋅ s p ± zα / 2 ⋅ p (1 − p ) n n 2 Confidence Interval for μ – σx known.
Lower Limit Confidence interval for μ - σx not known.
Lower Limit Point Estimate Estimate Upper Limit Point Estimate Upper Limit x − zα / 2 ⋅ σ x x
Critical values x + zα / 2 ⋅ σ x x − tα / 2 ⋅ sx n x
t-value that leaves area α/2 to the right x + tα / 2 ⋅ sx n E = zα / 2 ⋅ σ x = zα / 2 ⋅ σx
n Margin of Error. Estimate of the Standard Error Confidence Interval for π .
Lower Limit Point and Interval Estimation.
Upper Limit Point Estimate Estimate p − zα / 2 ⋅ σ p
Margin Margin of of Error. p
e = zα / 2 ⋅ p + zα / 2 ⋅ σ p
p (1 − p ) n Sample Sample Size: You want a certain margin of error given your chosen level of confidence: Solve Solve for n: 2 2 ⎛ zα / 2 ⋅ σ x ⎞ ( zα / 2 ) ⋅ σ x2 Or:
n=⎜ ⎝ e ⎟= ⎠ e2 CI for μ CI for π n= ( zα / 2 ) π
2 (1 − π ) e2 What’s the difference? 3 Hypothesis Hypothesis Tests. Single Population Parameter Hypothesis Tests:
Identify the Unknown Population Parameter: We We test “claims” about unknown population unknown parameters. Each Each hypothesis: Null and Alternative. Which Which is the “research question?” Must Must have a specific numeric value – which hypothesis is “equal to” that value? μx
σx known z-test σx not known t-test π
Always z-test X and zcalc X , s x and tcalc p and zcalc What’s required to insure normal? TwoTwo-Sample Hypothesis Tests:
Are the Two Population Parameters Equal? μ1 = μ2
σx1 and x2 not known t-test π1 = π2
Always z-test X 1 and X 2 std. error: combine sx1 and sx 2 tcalc p1 and p2 ˆ use p for standard error zcalc Six Six steps for any hypothesis test: Step 1: Hypothesis – null and alternative. Step 2: Choose α. 2: Choose Step 3: Determine critical value(s). Draw! critical Step 4: Select a sample, estimate and estimate calculate calculate the test statistic. Step 5: Compare – calculated test statistic to critical value. Place test stat. on picture. Step 6: Write conclusion. 4 ...
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- Spring '11