Hypothesis Testing for Population Means and Proportions

Topics

• Hypothesis testing for population means:– z test for the simple case (in last lecture)– z test for large samples– t test for small samples for normal distributions

• Hypothesis testing for population proportions:– z test for large samples

z-test for Large Sample Tests

• We have previously assumed that the population standard deviationσis known in the simple case.

• In general, we do not know the population standard deviation, so we estimate its value with the standard deviation s from an SRS of the population.

• When the sample size is large, the z tests are easily modified to yield valid test procedures without requiring either a normal population or known σ.

• The rule of thumb n > 40 will again be used to characterize a large sample size.

z-test for Large Sample Tests (Cont.)

• Test statistic:

• Rejection regions and P-values:– The same as in the simple case

• Determination of β and the necessary sample size:– Step I: Specifying a plausible value of σ

– Step II: Use the simple case formulas, plug in theσ estimation for step I.

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t-test for Small Sample Normal Distribution

• z-tests are justified for large sample tests by the fact that: A large n implies that the sample standard deviation s will be close toσfor most samples.

• For small samples, s and σare not that close any more. So z-tests are not valid any more.

• Let X1,…., Xn be a simple random sample from N(μ, σ). μ and σ are both unknown, andμ is the parameter of interest.

• The standardized variable

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The t Distribution

• Facts about the t distribution:

– Different distribution for different sample sizes

– Density curve for any t distribution is symmetric about 0 and bell-shaped

– Spread of the t distribution decreases as the degrees of freedom of the distribution increase

– Similar to the standard normal density curve, but t distribution has fatter tails

– Asymptotically, t distribution is indistinguishable from standard normal distribution

Table A.5 Critical Values for t Distributions

Degrees of Freedom 0.1 0.05 0.025 0.01 0.0051 3.078 6.314 12.706 31.821 63.6572 1.886 2.92 4.303 6.965 9.925. . . . . .. . . . . .

20 1.325 1.725 2.086 2.528 2.845. . . . . .. . . . . .

200 1.286 1.653 1.972 2.345 2.601z* 1.282 1.645 1.96 2.326 2.576

α = .05

t-test for Small Sample Normal Distribution (Cont.)

• To test the hypothesis H0:μ = μ0 based on an SRS of size n, compute t test statistic

• When H0 is true, the test statistic T has a t distribution with n -1 df.

• The rejection regions and P-values for the t tests can be obtained similarly as for the previous cases.

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Recap: Population Proportion

• Let p be the proportion of “successes” in a population. A random sample of size n is selected, and X is the number of “successes” in the sample.

• Suppose n is small relative to the population size, then X can be regarded as a binomial random variable with

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Recap: Population Proportion (Cont.)

• We use the sample proportion as an estimator of the population proportion.

• We have

• Hence is an unbiased estimator of the population proportion.

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Recap: Population Proportion (Cont.)

• When n is large, is approximately normal. Thus

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• We can use this z statistic to carry out hypotheses for

H0: p = p0 against one of the following alternative hypotheses:

– Ha: p > p0

– Ha: p < p0

– Ha: p ≠ p0

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Large Sample z-test for a Population Proportion

• The null hypothesis H0: p = p0

• The test statistic is

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Alternative Hypothesis

P-value Rejection Region for Level α Test

Ha: p > p0 P(Z ≥ z) z ≥ zα

Ha: p < p0 P(Z ≤ z) z ≤ - zα

Ha: p ≠ p0 2P(Z ≥ | z |) | z | ≥ zα/2

Determination of β

• To calculate the probability of a Type II error, suppose that H0 is not true and that p = p instead. Then Z still has approximately a normal distribution but

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• The probability of a Type II error can be computed by using the given mean and variance to standardize and then referring to the standard normal cdf.

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Determination of the Sample Size

• If it is desired that the level αtest also have β(p) = β for a specified value of β, this equation can be solved for the necessary n as in population mean tests.

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