Numeracy

Statistics: How to Calculate P-Value




The p-value is a statistical measure that helps determine the significance of results in hypothesis testing. Here's how to calculate it:

1. State the Hypotheses

  • Null Hypothesis (H?): Assumes no significant difference or effect (e.g., = 10).
  • Alternative Hypothesis (H?): Assumes a significant difference or effect (e.g., 10).

Example:
- H?: = 9 inches
- H?: 9 inches

2. Use the T-Test Formula

To calculate the test statistic ( t ), use the formula:

[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
]

Where:
- ( t ): Test statistic
- ( \bar{x} ): Sample mean
- ( \mu ): Hypothesized mean
- ( s ): Standard deviation of the sample
- ( n ): Sample size

Example Calculation:
- Sample mean (( \bar{x} )) = 8 inches
- Hypothesized mean (( \mu )) = 9 inches
- Standard deviation (( s )) = 2 inches
- Sample size (( n )) = 31

[
t = \frac{8 - 9}{\frac{2}{\sqrt{31}}} = \frac{-1}{0.35921} = -2.78388
]

Take the absolute value: ( |t| = 2.78388 ).

3. Use a T-Distribution Table to Find the P-Value

  • Degrees of Freedom (df): Subtract 1 from the sample size: ( df = n - 1 = 30 ).
  • Significance Level: Check a t-distribution table at the given level (e.g., 0.01).

Example:
- With ( df = 30 ) and ( |t| = 2.78388 ):
- ( |t| ) falls between values for 0.005 and 0.001.
- Average these values: ( (0.005 + 0.001) / 2 = 0.003 ).
- For a two-tailed test, multiply by 2: ( p = 0.003 * 2 = 0.006 ).

Result Interpretation

  • If ( p \leq \alpha ): Reject the null hypothesis (( H? )).
  • If ( p > \alpha ): Fail to reject ( H? ).

Example Outcome:
- ( p = 0.006 ), which is less than the 0.01 significance level.
- Reject ( H? ): The mean rainfall for August is not 9 inches.

Summing it up

  • Small p-value: Strong evidence against ( H? ).
  • Large p-value: Data aligns with ( H? ).
  • Use a two-tailed test unless you have a specific directional hypothesis.

By mastering p-value calculations, you can confidently assess statistical significance in data analysis!

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