Question

A researcher wants to know the average score in statistics of all students in public policy programs in the US. He surveys a random sample of 400 students and finds that the average score of these 400 students is 70 points. The sample standard deviation is 20 points. Assume that the population standard deviation equals the sample standard deviation. Find the z critical value for the 80% confidence interval (CI). Explain how you find this number.

Answer 1

To calculate the z critical value for an 80% confidence interval, we look for a z-score that represents the central 80% of the normal distribution, leaving 5% in each tail. This value is approximately 1.282.

Step-by-step explanation:

To find the z critical value for an 80% confidence interval, we need to determine the z-score that corresponds to the central 80% of the standard normal distribution. This means that we want to find the z-scores that leave 10% of the distribution's area in the tails (5% in each tail since the normal distribution is symmetrical).

Using a standard normal distribution table or a calculator that provides z-scores for given probabilities, we can locate the z-score that corresponds to the area to the left of the z-score being 0.90. This is because 0.90 represents the area to the left of the positive z critical value plus the 0.10 in the tails. Thus, the z critical value for an 80% confidence interval is approximately 1.282.

To explain this further with an example, if we have a sample mean of 70 and a sample standard deviation of 20, and we want an 80% confidence interval for the true mean, we multiply our z critical value by the sample standard deviation and then add and subtract this margin of error from the sample mean to get our confidence interval.