1. The average life expectancy of people around the world is 71.30 years with a population standard deviation of 8.00 years. I take a sample of 50 countries now and find that the average life expectancy is 74.37 years. Suppose I’m interested in seeing if there has been a significant increase in the life expectancy.
a. What is the standard error?
b. What is the margin of error at 90% confidence?
c. Using my sample of 50, what would be the 90% confidence interval for the population mean?
d. If I wanted to control my margin of error and set it to 1 at 95% confidence, what sample size would I need to take instead of the 50?
e. What are the null and alternative hypotheses?
f. What is the critical value at 90% confidence?
g. Calculate the test statistic.
h. Find the p-value.
i. What conclusion would be made here at the 90% confidence level?
a. The standard error measures the variability of the sample mean and is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the standard error can be calculated as follows:
Standard Error = (Population Standard Deviation) / √(Sample Size)
= 8 / √(50)
= 8 / 7.071
≈ 1.131
b. The margin of error represents the range within which the population mean is likely to fall. It is calculated by multiplying the standard error by the appropriate critical value. For a 90% confidence level, you would need to find the critical value (z-value) corresponding to a 5% significance level (since it is a two-tailed test). The margin of error can be calculated as follows:
Margin of Error = Critical Value * Standard Error
c. To calculate the 90% confidence interval for the population mean, you would use the formula:
Confidence Interval = Sample Mean ± Margin of Error
d. To calculate the required sample size for a desired margin of error, you would use the formula:
Sample Size = ((Z-value * Standard Deviation) / Margin of Error)^2
e. The null hypothesis would state that there has been no significant increase in the life expectancy, while the alternative hypothesis would state that there has been a significant increase.
f. The critical value for a 90% confidence level can be obtained from a standard normal distribution table or calculated using statistical software.
g. The test statistic is calculated by subtracting the population mean from the sample mean, and then dividing it by the standard error.
Test Statistic = (Sample Mean - Population Mean) / Standard Error
h. The p-value represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. It can be calculated using statistical software or by referring to a t-distribution table.
i. To make a conclusion at the 90% confidence level, you would compare the p-value to the chosen significance level (usually 0.05 or 0.01). If the p-value is less than the significance level, you would reject the null hypothesis and conclude that there is evidence of a significant increase in the life expectancy. Otherwise, if the p-value is greater than the significance level, you would fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant increase.