Summary of Formulae for One Sample Hypothesis Tests

Use the tool that you are most comfortable using. Formulae below use the following symbols: α = alpha = 1 – Confidence Level, σ = population standard deviation, s = sample standard deviation, μ = population mean, = sample mean, n = sample size, x = a data value, p is the percent below, df = degrees of freedom. Note z_α/2 is the z-score for percent α/2, normally converted to positive, similarly with t_α/2. 

General Steps for a One Sample Hypothesis Test

1. Determine the Null & Alternate Hypotheses

    a) H₀ will always contain one of the ‘equal’ symbols, =, ≤, or ≥. The main population parameters to test are mean, proportion, and sometimes, standard deviation.

    b) H_a will never contain an ‘equal’ symbol, ≠, <, or >.

2. Write a quality sentence describing what your random variable represents, it will be the sample variable that matches the population in 1a.

3. Determine which distribution you need, and the parameters you have for it.

    a) If you are working with the means and know the Population Standard Deviation, it will be the Normal Distribution. (see below for all calculations).

    b) If you are working with the means and don’t know the Population Standard Deviation, it will be the Student’s t-Distribution.

    c) If you are working with a proportion and have np’ > 5 & nq’ > 5, you will use the Normal Distribution for Proportions.

4. Find your test statistics & p-value.

5. Find the critical values (optional if using p-value).

6. Draw a graph with your rejection region and test statistic (optional, but can be very helpful).

7. Make a decision
a) State the α
b) Decision: reject or do not reject the null hypothesis
c) Reason for decision: either test statistic is in rejection region, p-value relation to α, etc
d) Conclusion: clearly write your conclusion using the null/alternate hypotheses and quality sentences.

Normal Distributions (Known Population Standard Deviation)

The population mean, µ will be what you are testing normally.

1. “By Hand”

     Standard deviation: \frac{\sigma_x}{\sqrt{n}}:

        

      Test Statistics: z=\frac{\overline{x}-\mu_x}{\sigma_x/\sqrt{n}}, this conversion is optional for most calculators.

      Critical Value: is the t-value at the Significance level, α, if left-tailed, 1 – α , if right tailed, and α/2, if two-tailed.

      p-value, is the probability of the Test Statistic being, less than, or greater than other values, depending on the type of test.

2. Desmos: Calculate Critical, test, and p-values

3. The “84 Calculators”: If you have the list, use STAT > TEST > line 1 Z-Test, then enter your μ, σ, your list of numbers (tedious to do if they aren’t already in there), Freq (usually 1), & Alternate Hypothesis.

         If you have the statistics , use STAT > TEST > line 1 Z-Test > click the Stats at the top, then enter your  μ, σ, , n, & & Alternate Hypothesis.

 4. Normal/t-distribution Probabilities website: https://s3-us-west-2.amazonaws.com/oervm/stats/probs.html

Student t-Distributions (Unknown Population Standard Deviation)

  Degrees of Freedom, df = n – 1, needed for all t calculations

1. “By Hand”

     Standard deviation: \frac{s}{\sqrt{n}}

      \overline{X}\:\sim\:t-Dist\left(degrees\:of\:freedom\right)

      Test Statistic: t=\frac{\overline{x}-\mu_x}{s/\sqrt{n}}, you need to convert the sample mean to the t-value, unless using # 4 below.

      Critical Value: is the t-value at the Significance level, α, if left-tailed, 1 – α , if right tailed, and α/2, if two-tailed.

      p-value, is the probability of the Test Statistic being, less than, or greater than other values, depending on the type of test.

2. Desmos: Calculate Critical, test, and p-values

3. “84 Calculators”: If you have the list, use STAT > TEST > line 2 T-Test, then enter your μ, your list of numbers (tedious to do if they aren’t already in there), Freq (usually 1), & Alternate Hypothesis.

          If you have the statistics , use STAT > TEST > line 2 T-Test > click the Stats at the top, then enter your  μ, , s, n, & & Alternate Hypothesis.

 4. Normal/t-distribution Probabilities: https://s3-us-west-2.amazonaws.com/oervm/stats/probs.html

Proportion Hypothesis Test in Parts (np > 5; nq > 5)

1. “By Hand”

     p’ = x/n, where x is the # ‘positive’, for the study and n is the sample size.

     p₀ = population proportion that you are testing; q₀ = 1 – p₀

     Determining the Test & the Hypotheses, similar to the others, for proportions, you do need np > 5; nq > 5

     Calculate z-test statistics: z=\frac{p'-p_0}{\sqrt{\frac{p_0q_0}{n}}} (see below for calculators)

     Find the p-value of the  test score (remember here p-value is the probability of test score happening!)

     Make a Decision: if p-value ≤ α, Reject the Null Hypothesis, if p-value > α, Fail to Reject the Null Hypothesis (NEVER “ACCEPT”)

     Conclusions, remember, this part is about your results, you are rejecting or not the Null Hypothesis and determining what the information tells you

2. Desmos: Calculate Critical, test, and p-values

     Desmos Link: https://www.desmos.com/calculator/yzmtbpddom

3. The “84 Calculators”: Stat > Test > Line 5 1-PropZTest(p0, x, n, alt), where values are as above, and alt is your alternate hypothesis, two-tailed, left-tailed, or right-tailed.

Just a reminder The “84 Calculators” are either the Texas Instruments TI-84, or the apps based on it: Calc84 (Graphing Calculator plus 84 graph emulator free 83) by lethinhien for Android and Graphing Calculator Plus by Incpt.Mobis for Apple.