H2 Maths 6.5 - One-Tailed Z-Test for Population Mean

Step 1 - State the hypotheses

A drinks manufacturer claims each can contains a mean of 330 millilitres.

Step 1 - State the hypotheses

A consumer group suspects the cans are underfilled, so they set up a hypothesis test.

Step 1 - State the hypotheses

The null hypothesis is that the population mean equals 330. The alternative hypothesis is that the mean is less than 330 - a lower one-tailed test.

Step 2 - Set up the test statistic

The sample size is 100, which is large.

Step 2 - Set up the test statistic

By the Central Limit Theorem, the sample mean is approximately normally distributed.

Step 2 - Set up the test statistic

We use a Z-test with the standard error equal to sigma over root n.

Step 3 - Calculate the test statistic

Subtract the hypothesised mean from the sample mean, then divide by the standard error.

Step 3 - Calculate the test statistic

328.5 minus 330 is negative 1.5. Dividing by 0.75 gives z equals negative 2.

Step 3 - Calculate the test statistic

The negative value confirms the sample mean lies below the claimed mean.

Step 4 - Find the p-value

For a lower one-tailed test, the p-value is the probability that Z is less than the calculated value.

Step 4 - Find the p-value

From standard normal tables, P of Z less than negative 2 equals 0.0228.

Step 4 - Find the p-value

Compare this to the significance level of 0.05.

Step 5 - State the conclusion

Since 0.0228 is less than 0.05, we reject H_0.

Step 5 - State the conclusion

There is sufficient evidence at the 5% significance level that the cans are being underfilled.

Step 5 - State the conclusion

Common pitfall: always write the conclusion in context - do not just say "reject H-naught" without referring to the original claim.