Questions: In a random sample of 6 residents of the state of California, the mean waste recycled per person per day was 2.7 pounds with a standard deviation of 0.74 pounds. Determine the 80% confidence interval for the mean waste recycled per person per day for the population of California. Assume the population is approximately normal. Step 1 of 2: Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

Solution Steps

To construct an 80% confidence interval for the mean waste recycled per person per day, we need to find the critical z-value that corresponds to the middle 80% of the standard normal distribution. This means we are looking for the z-score that leaves 10% in each tail of the distribution.

Using the cumulative distribution function $\Phi$, we can express the probability as follows:

$$P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.1) - \Phi(-\infty) = 0.54$$

From this calculation, we find that the critical z-value for the 80% confidence interval is:

$$Z_{end} = 0.1$$

The critical z-value that will be used in constructing the confidence interval is:

$$\text{Critical z-value} = 0.1$$

Final Answer