Question

A random sample of 50 smokers were questioned about the number of hours they sleep each day. The sample of smokers sleeps on average 7.5 hours per day with a standard deviation of 0.4 hours. We want to test the hypothesis that smokers need less sleep than non-smokers, who need an average of 7.7 hours of sleep. Perform the test at a 1% level of significance. Find the critical value.

Answer 1

To test the hypothesis that smokers need less sleep than non-smokers, we can perform a one-sample z-test for the mean. Here are the steps to perform the test:

**Step 1: State the Hypotheses**

- Null Hypothesis (\(H_0\)): The mean hours of sleep of smokers is equal to that of non-smokers.
\[ H_0: \mu = 7.7 \]

- Alternative Hypothesis (\(H_a\)): The mean hours of sleep of smokers is less than that of non-smokers.
\[ H_a: \mu < 7.7 \]

**Step 2: Calculate the Test Statistic**

We will use the formula for the z-test statistic:

\[
z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 7.5\) (sample mean)
- \(\mu_0 = 7.7\) (hypothesized population mean)
- \(s = 0.4\) (sample standard deviation)
- \(n = 50\) (sample size)

Now, substituting the values into the formula:

\[
z = \frac{7.5 - 7.7}{\frac{0.4}{\sqrt{50}}}
\]

First, let’s calculate the denominator:

\[
\frac{0.4}{\sqrt{50}} \approx \frac{0.4}{7.0711} \approx 0.05657
\]

Now, calculate the z value:

\[
z = \frac{7.5 - 7.7}{0.05657} \approx \frac{-0.2}{0.05657} \approx -3.53
\]

**Step 3: Determine the Critical Value**

At a 1% significance level (\(\alpha = 0.01\)) for a one-tailed test, we will find the critical value from the standard normal distribution table.

Using a Z-table or a calculator, the critical z value for \(\alpha = 0.01\) (one-tailed) is approximately:

\[
z_{critical} = -2.33
\]

**Step 4: Make a Decision**

Now, we compare our calculated z value with the critical z value.

- Calculated z: \(-3.53\)
- Critical z: \(-2.33\)

Since \(-3.53 < -2.33\), we reject the null hypothesis.

**Conclusion**

At the 1% significance level, there is sufficient evidence to conclude that smokers need less sleep than non-smokers. The critical value for this test is \(-2.33\).