The p-value provided by your technology determines your conclusion:
- If p-value < 0.01, you would conclude "Reject the null hypothesis."
- If p-value ≥ 0.01, you would conclude "Do not reject the null hypothesis."
To find the P-value using a t-test, we can follow these steps:
Look up the P-value in a t-distribution table or use a calculator.
Given the information:
Sample size (n): 33
Sample mean (X): 5.67
Population mean (µ): 5.67 (given)
Test statistic (t): -2.625
To find the p-value for this hypothesis test, you would follow these steps:
Step 1: State the Null and Alternative Hypotheses
The null hypothesis (H0): The mean weight of quarters \( \mu \) is equal to
5.670 grams.
The alternative hypothesis (H1): The mean weight of quarters \( \mu \) is
not equal to 5.670 grams.
This is mathematically represented as:
H0: \( \mu = 5.670 \) g
H1: \( \mu \\eq 5.670 \) g
Step 2: Determine the Degrees of Freedom (df)
For a Student's t-test, the degrees of freedom are calculated as the
sample size minus one. In this case, with a sample size \( n \) of 33:
\[ \text{df} = n - 1 = 33 - 1 = 32 \]
Step 3: Calculate the P-value
You have the test statistic \( t \), which is -2.625, for a sample size of 33,
which gives you 32 degrees of freedom as calculated above.
The p-value for the t-test can be found using statistical software, a
calculator with statistical functions, or a t-distribution table. Since this is a
two-tailed test (because the alternative hypothesis does not specify the
direction of the difference), the p-value must account for the probability
of observing a test statistic as extreme as -2.625 or lower, as well as the
same extreme in the positive direction.
Using a t-distribution, the p-value is found by looking up the cumulative
probability of the test statistic and then doubling it (for a two-tailed test).
Step 4: Draw a Conclusion
Compare the p-value with the significance level of 0.01. If the p-value is
less than 0.01, you would reject the null hypothesis. If the p-value is
greater than or equal to 0.01, you would not reject the null hypothesis.
For example, if the technology you are using gives you a p-value of 0.012
(for instance), this would be your step-by-step process:
1. Calculate degrees of freedom: \( \text{df} = 33 - 1 = 32 \).
2. Use technology to find the p-value for a t-statistic of -2.625 with 32
degrees of freedom.
3. Double the one-tailed p-value to account for the two-tailed test.
The p-value provided by your technology determines your conclusion:
- If p-value < 0.01, you would conclude "Reject the null hypothesis."
- If p-value ≥ 0.01, you would conclude "Do not reject the null hypothesis."