One-Sample Z-Test for a Population Mean
Perform a one-sample z-test to determine whether a population mean differs from a hypothesized value when the population standard deviation is known.
Problem Scenario
A cereal manufacturer claims that each box contains an average of 500 grams of cereal. A quality control analyst suspects that the filling machine is underfilling boxes. She randomly selects 36 boxes and finds a sample mean of 496.2 grams. From years of production data, the population standard deviation is known to be 12 grams. Test at the 0.05 significance level whether the mean weight is less than 500 grams.
Given Data
Requirements
- State the null and alternative hypotheses
- Calculate the z-test statistic and find the p-value
- Make a decision and state the conclusion in context
Solution
Step 1:
State the hypotheses. H_0: mu = 500 (the mean filling weight is 500 grams). H_a: mu < 500 (the mean filling weight is less than 500 grams). This is a left-tailed test.
Step 2:
Calculate the standard error: SE = sigma / sqrt(n) = 12 / sqrt(36) = 12 / 6 = 2.0 grams.
Step 3:
Calculate the z-test statistic: z = (x-bar - mu_0) / SE = (496.2 - 500) / 2.0 = -3.8 / 2.0 = -1.90.
Step 4:
Find the p-value. For a left-tailed test, p-value = P(Z < -1.90). Using the standard normal table, P(Z < -1.90) = 0.0287.
Step 5:
Decision: Since the p-value (0.0287) is less than alpha (0.05), we reject the null hypothesis. There is sufficient evidence at the 0.05 level to conclude that the mean filling weight is less than 500 grams.
Final Answer
z = -1.90, p-value = 0.0287. At the 0.05 significance level, we reject H_0. There is statistically significant evidence that the cereal boxes are being underfilled (mean weight is less than 500 grams).
Key Takeaways
- ✓A z-test is appropriate when the population standard deviation is known and the sample size is large enough (n >= 30) for the Central Limit Theorem to apply.
- ✓The p-value gives the probability of observing a result as extreme as (or more extreme than) the sample result, assuming the null hypothesis is true.
- ✓Rejecting H_0 does not prove H_a is true with certainty; it means the evidence against H_0 is strong enough at the chosen significance level.
Common Errors to Avoid
- ✗Using a two-tailed test when the research question specifies a direction. Read the problem carefully to determine whether the test is one-tailed or two-tailed.
- ✗Confusing the population standard deviation (sigma) with the standard error (sigma / sqrt(n)). The standard error accounts for sample size and is always smaller than sigma.
- ✗Stating the conclusion in purely statistical terms without relating it back to the context of the problem.
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Common questions about this problem type
Use a z-test when the population standard deviation (sigma) is known. Use a t-test when sigma is unknown and you must estimate it from the sample using s. In practice, sigma is rarely known, so t-tests are far more common.
Convention varies, but most textbooks treat p-value = alpha as borderline and still reject H_0 (using the strict inequality p <= alpha for rejection). In practice, you should report the exact p-value and let the reader judge significance.