L8 Test statistics comparing 2 populations

Purpose of the comparison

- Comparison of two populations using hypothesis tests and confidence intervals. The comparison generally seeks to determine if two averages (μ) or two proportions (π) are equal.

Examples of comparison:

- Determining if the mean sales of a product are equal before (μb​) and after (μa​) advertisement

- Comparing the mean incomes in Sweden (μs​) and Finland (μf​)

- Comparing the proportion of bankrupt firms in Sweden (πs​) and Denmark (πd​)

Two tailed or one tailed:

If your goal is to see whether two averages or proportions are equal, that's usually a two-tailed test.

But if you have a directional claim (e.g., "Group A performs better than Group B"), then a one-tailed test might be appropriate.

Two general research strategies are employed for comparing populations:

1. Matched Samples Design

2. Independent Samples Design

Matched Samples: (also called matched pairs, paired samples, or dependent samples)

are established by pairing participants so they share every characteristic except the one under investigation.

Matched samples characteristics

- Purpose: Improve statistics by controlling for the effects of other "unwanted" effects/variables.

- A participant can be a person, firm, or object.

- Pairs can be different individuals assigned to a treatment group and a control group.

- Pairs can also be the same individual measured at different times (e.g., before and after treatment)

Categories of matched samples

- Same individuals/objects measured at different times

o Treatment: before and after e.g. before sales and after sales.

o To make inferences use:

§ Hypothesis test

§ Confidence interval for difference in means

§ Effect size (d): Difference between before and after

- Different individuals/objects measure for two different "treatments"

o Treatment 1 and 2 e.g. one ad and another ad.

Common use of matched samples

- A common use is assigning one individual to a treatment group (given a new drug) and a similar individual to a control group (given a sugar pill - no changes). The treatment group is the one changed.

Highlights: for the d: the difference within each pair (di​).

- This converts the two-sample problem into a one-sample test focused on the mean of the differences (μd​).

- If the test statistic falls outside the critical range (e.g., 2.13 is greater than the upper critical value), the null hypothesis (H0​) is rejected.

Calculation - matched samples

1. Step 1: Calculate variable d - difference between before and after .

2. Step 2: Add all observations and divide with mean - formula from 1 or 2nd lecture.

3. Step 3: Calculate standard deviation - take square root of variance.

4. Step 4: Calculate test statistics and since it's t distribution and calculate df.

5. Step 5: 2 tailed or one tailed.

6. Step 6: Find the critical values by looking at t distribution.

1. Step 7: Look if we will reject the hypothesis by putting the t in the distribution.

2. Step 8: We will reject null hypothesis since 2.13 is greater than upper critical value. Reject because it's not equal but lower absent days and seems to be working.

3. Step 6.2: P value. Search for 2.13 in the t distribution where df is. Probability between 5 and 2,5 %. Reject since p-value is less than a.

Calculate using P-value: Calculate probability of gaining 2.13 or more extremes e.g. higher than 2.13 or smaller than 2.13.

Normally distributed

means it should be above 30 for central theorem or mean that it is applicable.

Calculate Confidence interval - matched samples

- Use formula and find t distribution in the t formula to find the critical value.

- P value - further away from the null hypothesis. Find 2.8 which is the t in the t table. See that its smaller than 2.8 and smaller than 0.005. p-value

Calculation: Two independent samples:

Take one sample from each population, one and from the second one. No relationships between two samples. Sample size do not need to match either.

1. Take sample and calculate mean, standard deviation and number of observation.

2. Calculate t using formula while df formula is more complicated. Round downwards if df is not an integer. If round up you push the point out of the tail which isn't preferred. Round up then higher value.

Example 1

1. Define hypothesis - one or two tailed.

2. Calculate test statistics. Take the two samples into the formula.

3. Get the value of the test statistics and calculate the degrees of freedom and round down.

4. Do critical value and realize ttwo sided or one sided hypothesis. Search for the points by looking at the t values and write the points.

5. P-value find the t value in the t distribution and look up in the area and see where the t value is between the two types of p-values on the over line.

6. Get a positive value between the means in the test statistics shows that the first have a higher salary.

Highlights

Confidence interval

Remember have at least 30 observations.

P-value

1. Realize if have one sided or two sided test by looking at the alternative, it says smaller than then so p-value is an even smaller one

Matched samples have similar sample sizes or have before and after while independent samples don't need similar sample sizes.

Calculation of proportions: Test hypothesis using proportions with 2 independent samples.

1. Use the formula for test statistics. Check so limit theorem where the conditions are above 5 for proportion formula and for the confidence interval, MUST CHECK EVERYTIME.

2. Estimate the proportions by taking the sample divided by sample sizes.

3. Now do the test hypothesis

4. Calculate the test statistics but first calculate p to apply.