Scenario 1: You Are Given Raw Data (A Ratio Variable) and Asked to Find a 95% Confidence

Scenario 1: You are given raw data (a ratio variable) and asked to find a 95% confidence interval.

Step 1. Find the mean of the data:

Step 2. Find the standard deviation using this formula

Step 3. Find the standard error using this formula

SE =

Step 4. Find the confidence interval using this formula

Lower Bound: - 1.96 (SE)

Upper Bound: + 1.96 (SE)

Step 5: Write your confidence interval as follows (Lower bound, Upper Bound)

Scenario 2. You are given 2 sets of data (both ratio variables) and asked to test for a difference of means using the confidence interval method

Step 1. State the null hypothesis (H0) and Alternative hypothesis (HA) as follows

H0: : µ1- µ2 = 0

HA: µ1- µ2 ≠ 0

Step 2. Find the means of each data set. Label as 1 and 2 accordingly

Step 3. Find the variance of each set of data (individually) using this formula. This values ares21 and s22accordingly (this is because s, standard deviation, is the square root of variance. Thus, s2is the variance)

s2 =

Step 4. Find the pooled standard error using this formula

Pooled SE=

Step 5. Find the confidence interval for the difference of means using this formula and compute algebraically

Lower Bound: 1 - 2 - 1.96(Pooled SE)

Upper Bound: 1 - 2 + 1.96(Pooled SE)

Write the confidence interval as (Lower Bound, Upper Bound)

Step 6. If zero is within your confidence interval (meaning if falls above the lower bound and below the upper bound), then you DO NOT reject the null hypothesis. If zero is not within the confidence interval, you may reject the null hypothesis.

NOTE: If you are asked to do a difference of means test using the confidence interval method and have already been given the means and standard deviations (or variances), then you only need to do steps 1 and 4-6.

Scenario 3: You are given 2 sets of data and asked to find the difference of means using the t-statistic (t-test) method for a two-tailed test.

Step 1. State the null hypothesis that the means are the same (H0) and Alternative hypothesis (H1) that the means are different as follows

H0: : µ1- µ2 = 0

HA: µ1- µ2 ≠ 0

Step 2. Find the means of each data set. Label as 1 and 2 accordingly

Step 3. Find the variance of each set of data (individually) using this formula. This values are s21 and s22 accordingly (this is because s, standard deviation, is the square root of variance. Thus, s2is the variance)

s2 =

Step 4. Find the t-statistic using this formula:

Step 5. If the t-statistic is greater than 1.96 or less than -1.96, than you can reject the null hypothesis. This means that the means are in fact different.

Scenario 4: You are given 1 set of data and asked to test the hypothesis of whether the mean is different than zero (a two-tailed test)

Step 1: State the null and alternative hypotheses as follows:

H0: µ= 0

HA: µ≠ 0

Step 2: Find the mean

Step 3: Find the standard deviation using the following formula

Step 4: Find the standard error using the following formula

SE =

Step 5. Find the t-statistic using the following formula:

Step 6. If the t-statistic is greater than 1.96 or less than -1.96, then you may reject the null hypothesis with 95% confidence.

NOTATION GUIDE:

: Sample Mean

µ : Population Mean (the true mean you are approximating)

s : Standard Deviation

s2: Variance

SE: Standard Error

N: Number in your sample

T: T-statistic