Probability Distributions and Hypothesis Tests: A Calc Pro Guide

Statistics classes are full of lookup tables: z-tables, t-tables, chi-square tables. Calc Pro's Statistics calculator replaces them with three tools, Distributions, Confidence Interval and Hypothesis Test, that give you exact probabilities and p-values in seconds.

Open the Statistics calculator and choose the tool from the list on the right. In the iPhone and iPad app these tools are part of Pro Premium; they're free in the web version.

Probability Distributions

Normal

Enter the Mean, Std Dev and an X Value.

Example: IQ scores have a mean of 100 and a standard deviation of 15. What share of people score 130 or less?

• z-Score: 2
• P(X ≤ x): 0.97725, so about 97.7% score 130 or less
• P(X ≥ x): 0.02275, so about 2.3% score 130 or more

Inverse Normal

Work backwards from a probability to a value.

Example: what IQ score marks the top 10%? Enter a mean of 100, a standard deviation of 15 and a probability of 90%. The answer is an x value of 119.22.

Binomial

For a fixed number of yes/no trials. Enter the Trials (n), the Success Prob. and the Successes (k).

Example: flip a fair coin 10 times. What's the chance of exactly 7 heads?

• P(X = k): 0.117188, about 11.7%
• P(X ≤ k): 0.945312, the chance of 7 or fewer heads
• P(X ≥ k): 0.171875, the chance of 7 or more heads

Poisson

For counts of events over time or space. Enter the Mean (λ) and the Events (k).

Example: a help desk gets 4 calls an hour on average. What's the chance of exactly 2 calls in the next hour?

• P(X = k): 0.146525, about 14.7%
• P(X ≤ k): 0.238103, 2 calls or fewer
• P(X ≥ k): 0.908422, 2 calls or more

t-Dist

Enter the Deg. of Freedom and a t Value.

Example: with 10 degrees of freedom and t = 2.228:

• P(T ≤ t): 0.974994
• P(T ≥ t): 0.025006
• Two-Tail p: 0.050012, which is why 2.228 is the familiar 5% critical value for 10 degrees of freedom

Chi-Square

Enter the Deg. of Freedom and a χ² Value.

Example: with 5 degrees of freedom and χ² = 11.07:

• P(X ≤ x): 0.94999
• P(X ≥ x): 0.05001, right at the 5% significance level

Confidence Intervals

Estimate a population mean from a sample. Choose Z (σ known) when you know the population standard deviation, or t (σ unknown) when you only have the sample's.

Example: a sample of 25 batteries lasts an average of 52 hours, with a standard deviation of 8 hours. Find a 95% confidence interval using t:

• Critical Value: 2.063899
• Margin of Error: 3.302238
• Lower Bound: 48.697762
• Upper Bound: 55.302238

You can be 95% confident that the true average battery life is between about 48.7 and 55.3 hours.

Hypothesis Tests

Test a claim about a mean. Choose 1-Samp Z, 1-Samp t or 2-Samp t.

One-Sample t-Test

Example: the manufacturer claims the batteries last 50 hours. Your sample of 25 averages 52 hours, with a standard deviation of 8. Is the difference significant?

Enter a Sample Mean of 52, a Std Dev of 8, a Sample Size of 25 and a Hypothesized Mean of 50:

• t Statistic: 1.25
• Deg. of Freedom: 24
• Two-Tail p: 0.223351
• One-Tail p: 0.111676

With p = 0.22, well above 0.05, the data doesn't show that the batteries last longer than claimed. The difference could easily be chance.

Two-Sample t-Test

Compare the means of two groups. Calc Pro uses Welch's t-test, which doesn't assume the two groups have the same spread.

Example: class A (30 students) averages 78 with a standard deviation of 10. Class B (28 students) averages 72 with a standard deviation of 12. Is the difference significant?

• t Statistic: 2.060871
• Deg. of Freedom: 52.72
• Two-Tail p: 0.044264
• One-Tail p: 0.022132

With p = 0.044, below 0.05, the difference between the classes is statistically significant at the 5% level.

Tips

1. Pick one or two tails before you test. Use two-tail p unless you predicted the direction of the difference in advance.
2. Use t when σ is unknown. That's almost always the case with real data.
3. Significant isn't the same as important. A tiny difference can be significant with a big enough sample; check whether it matters in practice.
4. Save your work. With Pro Premium, e-mail or save your results as a PDF.

For regression, descriptive statistics and data entry, see the Statistics Calculator guide.

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