Comparing Population Proportions and Means

• This chapter is about hypothesis tests comparing two populations.

Comparing Two Population Means

• Compares two population means with independent samples and known population standard deviations (σ₁ and σ₂).
• The goal is to examine the difference between two means.
• The sampling distribution shows results from subtracting the sampling distribution for the mean of one population from the sampling distribution for the mean of a second population.
• Formula for the Mean of the Sampling Distribution for the Difference in Means: μx̄₁-x̄₂ = μx̄₁ - μx̄₂
• μx̄₁ is the mean of the sampling distribution from Population 1.
• μx̄₂ is the mean of the sampling distribution from Population 2.
• The standard error of the difference between two means describes the variation in the difference between two sample means.
• The formula for the Standard Error of the Difference Between Two Means: σx̄₁-x̄₂ = √(σ₁²/n₁) + (σ₂²/n₂)
• σ₁ and σ₂ are the standard deviations for Populations 1 and 2.
• n₁ and n₂ are the sample sizes from Populations 1 and 2.
• If the sample size is small (n < 30), the hypothesis test requires that the population follow the normal distribution.
• If the sample size is large (n ≥ 30), from the Central Limit Theorem, the sampling distribution follows the normal distribution, so there are no restrictions on the population distribution.

Conducting a Hypothesis Test to Compare the Difference Between Two Means

• Identify the null and alternative hypotheses.
• Set a value for the significance level, a.
• Calculate the appropriate test statistic using the z-test
• The formula for the z-test statistic for a hypothesis test for the difference between two means (σ₁ and σ₂ known): z = (x̄₁ - x̄₂) - (μ₁ - μ₂)H₀ / σx̄₁-x̄₂
• (μ₁ - μ₂)H₀ is the hypothesized difference in population means.
• σx̄₁-x̄₂ is the standard error of the difference between two means.
• x̄₁ - x̄₂ is the difference in sample means between Populations 1 and 2.
• Determine the appropriate critical value using a z-score; the critical z-score identifies the rejection region for this two-tail test.
• Compare the z-test statistic z with the critical z-score zα/2. For a two-tail test, reject the null hypothesis if |z| > zα/2.
• Calculate the p-value.
• State the conclusion.

Using a Confidence Interval to Compare the Difference Between Two Means

• It helps to develop an interval estimate for the true difference in the population means.
• Assume that both sample sizes are 30 or more, otherwise the populations are normally distributed
• Formulas for the Confidence Interval for the Difference in the Means of Two Independent Populations: UCLx̄₁-x̄₂ = (x̄₁ - x̄₂) + zα/2σx̄₁-x̄₂ and LCLx̄₁-x̄₂ = (x̄₁ - x̄₂) - zα/2σx̄₁-x̄₂

Comparing Two Population Means with Unknown Population Standard Deviations

• Use sample standard deviations (s₁ and s₂) in place of population standard deviations (σ₁ and σ₂).
• The Student's t-distribution is to identify the rejection region.
• Both populations need to be normally distributed unless both sample sizes are 30 or larger.
• Case 1: The population variances are equal (σ₁² = σ₂²).
• Case 2: The population variances are not equal (σ₁² ≠ σ₂²).

Hypothesis Testing Procedure

• The hypothesis testing procedure when σ₁ and σ₂ are unknown but equal follows the same steps as when σ₁ and σ₂ are known, except for these differences
• The test statistic is a t-test statistic.
• The critical value is a t-score with df = n₁ + n₂ – 2.

Pooled Variance

• A pooled variance, the weighted average of the two sample variances, is calculated to estimate the unknown population standard deviation. s²p = [(n₁ - 1)s₁² + (n₂ - 1)s₂²] / [(n₁ + 1) + (n₂ - 1)]
• s₁² is the variance of the sample from Population 1.
• s₂² is the variance of the sample from Population 2.
• n₁ and n₂ are the sample sizes from Populations 1 and 2.
• The formula for the t-test Statistic for a Hypothesis Test for the Difference Between Two Means (σ₁ and σ₂ Unknown but Equal): tx̄ = [(x̄₁ - x̄₂) - (μ₁ - μ₂)H₀] / [sₚ√(1/n₁ + 1/n₂)], where:
• (μ₁ - μ₂)H₀ = The hypothesized difference in the population means
• s²ₚ = The pooled variance
• Use the t-distribution table for n₁ + n₂ – 2 degrees of freedom to determmine the appropriate critical value
• Compare the t-test statistic (tx̄) with the critical t-score: tα (if a one-tail test) or tα/2 (if a two-tail test).
• Approximate the p-value. A precise p cannot be determined since only selected columns are shown in the table.
• Estimate the p by finding the critical t that bracket the calculated value of tx̄.
• Remember that the p is a two-tail probability for two-tail tests but a one-tail probability for one-tail tests.
• State the conclusion. Consider Excel to run hypothesis tests and assume equal population variances
• If population variances are unequal, a pooled variance is not appropriate.

Testing Differences Other Than Zero

• Sometimes one wants to test whether the difference between means exceeds a certain value.
• Use the desired difference in the formula for the calculated test statistic instead of zero
• Hypothesis examples include: Ho: μ1 - μ2 ≤ $3.00 (Bills for slow music average $3.00 or less than fast) and H₁ : μ1 - μ2 > $3.00 (Bills for slow music average $3.00 more than fast)

Hypothesis Testing with Dependent Samples

• With dependent samples, each observation from one sample is related to an observation from the other sample, otherwise known as a matched-pair test
• Credit card balances of selected customers before and after a bank promotion
• Cholesterol levels of patients before and after medication is prescribed.
• After finding matched-pair differences, keep track of he positive and negative values for the matched-pair differences
• Formula for the Matched-Pair Difference: d = x₁ - x₂
• d = The matched-pair difference
• x₁ and x₂ = The matched-pair values from Populations 1 and 2, respectively
• Formula for the Mean of the Matched-Pair Differences: d = (Σdᵢ) / n
• d = The mean of the matched-pair differences
• dᵢ = The ith matched-pair difference
• n = The number of matched-pair differences
• The negative value for the average means that the average IQ score after the modules was higher than the average before the modules.
• Formulas for the Standard Deviation of the Matched-Pair Differences: s =√[Σ(d;-d)²]/n-1]
• Identify the null and alternative hypotheses. (IQ after the modules is less than or equal to before the modules)
• μα is the population mean matched-pair difference, defined to be the value before the modules minus the value after the modules.
• To find the test, the formula is: (μd) Ho = The population matched-pair difference from the null hypothesis Set a value for the significance level, and calculate the appropriate test statistic.
• The t-test statistic is: t = d-/Sn) Ho The critical The critical value is a t-score The critical value is a t-score -ta = -1.895 α = 0.05 reject Ho if and here .
• 1.23 isn't less than -1.895, therefore, reject Ho.
• By using Table 5 in Appenix A find | . The critical value is -4, or 2.
• Because the critical point, we cannot the null hypothesis for the alternative that

Comparing Two Population Proportions with Independent Samples

• They are used for hypothesis tests to compare two population proportions.
• Two populations are Population 1 and Population 2.
• p₁ = The true proportion of interest from Population 1
• p₂ = The true proportion of interest from Population 2
• p̄₁ = sample proportion from Population 1
• p̄₂ = sample proportion from Population 2
• n₁ = The sample size from Population 1
• n₂ = The sample size from Population 2
• The sampling distribution is the result of subtracting one sampling distribution for the proportion from a second sampling distribution.
• This represents of the difference between population proportions and sample proportions: Formula is: σp₁-p₂ = √( * = *) Formula for the Standard Error for the Difference in Population Proportions: approximated by a distribution if = 5
• Formulas for the Confidence Interval a + + 1-tailed: 0 Ha≠ 0 for testing follows earlier but we the difference between population and a hypothesis * + formula where = (Σ)/.9049) - .649)=

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Explore comparing two population proportions using independent samples, focusing on the meaning of ( p_1 ), standard error significance, and confidence intervals. Also covers comparing two population means with independent samples and known population standard deviations, including hypothesis testing.