State the null and alternative in terms of the population parameter p, never the sample statistic.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Three conditions
Random sampling, independence with the 10% condition, and a large enough sample so np₀ and n(1 − p₀) both reach 10.
Use the null value
Check the large-sample condition and compute the standard error using p₀ from the null, not the sample proportion.
The test statistic
z is the sample proportion minus p₀, divided by the standard error. It counts standard errors from the null value.
The p-value
It is the probability of a result at least this extreme, assuming the null is true. Small means surprising.
Conclude in context
Compare the p-value to alpha, then state the conclusion about the original claim, in context, with the p-value quoted.
The logic of a significance test
Assume the null hypothesis, compute how unusual the observed data would be under that assumption, and decide whether that is surprising enough to abandon it. The reasoning is proof by contradiction, softened by probability.
Hypotheses are about parameters
The null and alternative concern the population parameter, never the sample statistic. Writing hypotheses about the sample proportion is a specific error, since the sample value is already known.
What a p-value is
The probability of getting data at least as extreme as observed, if the null hypothesis were true. It is not the probability that the null is true, and it never has been.
Stating the conclusion
Compare the p-value with alpha, decide whether to reject, then state what that means about the original question in context. All three steps are graded, and the third is the one most often skipped.
Step 2: Try It Yourself
Tap and try it out.
Observed
Expected
Observed has the most. It has 12 more than Expected.
Step 3: Watch an Example
One step at a time.
Watch Kofi Test a Coin
A coin gives 62 heads in 100 tosses, and Kofi tests whether it is fair.
- Step 1
The hypotheses are p = 0.5 against p not equal to 0.5.
Step 4: Your Turn
Practice makes it stick.
The Standard Error
Problem 1 of 2
p₀ = 0.5 and n = 100. What is the standard error?
The Statistic
Problem 2 of 2
Sample proportion 0.62, p₀ = 0.5, standard error 0.05. What is z?
Run the Test
1 of 8
p₀ = 0.5, n = 400. Standard error?
2 of 8
p₀ = 0.5, n = 100. What is np₀?
3 of 8
Both np₀ and n(1−p₀) must be at least what number?
4 of 8
Sample 0.6, p₀ = 0.5, standard error 0.05. What is z?
5 of 8
p-value 0.016 at alpha 0.05. Reject the null? 1 yes, 0 no.
6 of 8
Should hypotheses be about the parameter or the statistic? 1 parameter, 2 statistic.
7 of 8
Put the test procedure in order.
- 1Check the random, independence and large-sample conditions.
- 2Compute the test statistic and the p-value.
- 3Compare to alpha and conclude in context.
- 4State hypotheses about the population proportion.
8 of 8
p₀ = 0.5, n = 2500. Standard error?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Sample 0.55, p₀ = 0.5, standard error 0.05. What is z?
Question 2 of 2
Which proportion is used to compute the standard error in a test?
What You Learned
- State hypotheses about the population proportion.
- Check random, independence and large-sample conditions using p₀.
- Compare the p-value to alpha and conclude in context.