Question

According to the historical data, the life expectancy in the United States is less than the life expectancy in Denmark. A new study has been made to see whether this has changed. Records of 280 individuals from the United States who died recently are selected at random. The 280 individuals lived a mean of 77.5 years with a standard deviation of 7.7 years. Records of 300 individuals from Denmark who died recently are selected at random and independently. The 300 individuals lived a mean of 76.1 years with a standard deviation of 6.8 years. Assume that the population standard deviations of lifetimes in these two countries can be estimated by the sample standard deviations, as the samples used were quite large. Construct a 90% confidence interval for H-H2, the difference between the life expectancy H1 in the United States and the life expectancy u2 in Denmark. Then complete the table below. Carry your intermediate computations to at least three decimal places. Round your answers to at least two decimal places. (If necessary, consult a list of formulas.) What is the lower limit of the 90% confidence interval? what is the upper limit of the 90% confidence interval?

Answer 1

The 90% confidence interval for the difference between the life expectancy in the United States and Denmark is calculated using the sample means, standard deviations, and sample sizes. After computations, the lower limit is 0.40 years and the upper limit is 2.40 years.

Step-by-step explanation:

To construct a 90% confidence interval for the difference between the life expectancies in the United States and Denmark, we use the formula for the confidence interval for the difference between two independent means:
CI = (μ1 - μ2) ± Z * √(σ²_1/n1 + σ²_2/n2)
Here, μ1 is the mean life expectancy in the United States, μ2 is the mean life expectancy in Denmark, σ^2_1 is the variance in the United States, σ^2_2 is the variance in Denmark, n1 is the sample size for the United States, and n2 is the sample size for Denmark.

Using the given data:

μ1 = 77.5
σ²1 = 7.7
n1 = 280
μ2 = 76.1
σ²2 = 6.8
n2 = 300

First, we calculate the standard error of the difference between the two means:

SE = √(7.7²/280 + 6.8²/300) = √(0.212 + 0.155) = √(0.367) = 0.606

For a 90% confidence interval, Z is approximately 1.645 since we are dealing with a normal distribution. Now we can calculate the margin of error (E):

E = Z * SE = 1.645 * 0.606 = 0.997

The difference between the two means, μ1 - μ2, is 77.5 - 76.1 = 1.4.

Now we calculate the confidence interval:

CI = 1.4 ± 0.997 = (0.403, 2.397)

The lower limit of the 90% confidence interval is 0.40 years and the upper limit is 2.40 years. This means that we are 90% confident that the true difference in life expectancy between the two countries lies within this range.