Question

1. Between 1900 and 2000, then mean age at which men in the US (who marry) get married for the first time had a mean of 24.8 years. The historical and current standard deviation of age of first marriage is 2.9 years. Since 2000, for a sample of 96 men, the mean age of marriage is 25.6 years. Find the 95% Confidence interval for the mean age men marry since 2000. a. What does the confidence interval tell us about the mean age of first marriage over time?

Answer 1

Before answering the question we need to identify the type of distribution the sample belongs to. To identify that lets look at the information available to us.

n = 96 ,sample size

mean of the population = 24.8

mean of the sample = 25.6

standard deviation of the population = 2.9

confidence interval = 95%

Since the sample size is greater than 30 and standard deviation of the population is known we can use normal distribution.

To find 95% confidence interval the following formula can be used.

te Confidence internal foromula mt omez cohere m- sample mean 2- value depends on land of confidence om - Standard essor of mean 5m= 0 wobese of population, 6 - Standard deviation no size of the sample

calculation of standard error and z value value is given in the following figure

Do size of the sample m-25.6 om= 2.a = 0:30 for & is as to confidence interval 20.02821-96 5%, with 2.5000 in each tail.

calculation of z value is as follows

matarsity 0.5 0.5 rore 0.415 O.HTS d=25% Of=25% 2=-1.96 95% 21.96 confidence interval

consider the above normal distribution. Since normal distribution is symmetrical, it can be divided into two parts of 0.5 each as shown in the above image. 95% confidence interval mean, 5% level if significance. Since it is two tailed test, there will be 2.5% (5%/2) alpha value on each tail.

The shaded region depicts alpha value of 2.5% which equals 0.025. consider the right side of tail. To find area of white region simply subtract 0.025 from 0.5, which gives the value 0.475. Now take z table look for 0.475 in table of values which should give the z value as 1.96.

Now plug these values on the formula for confidence interval. Look the following image.

Loworo confidence limit ന - 6 - Y 2 കട - 6 - 9. 30 x | 46 - 4s o\ confidence limit apper ന + 6 - 72 = Qs , + Gം 7 | 06 - 0 | ജിത് ത trume pocou ഷ ന്നി . അഹമ വർ വീ ീഷയ ട കേന്ദ് ചീഫ് . ഷെഫ ല. പടിക ം അണ് olSoco C

95% confidence interval for mean age of men marry since 2000 is 25.01 and 26.19.

b) A confidence interval is a range of values in which actual value is expected to lie. The 95% confidence interval for the above question means that there is a 95% probability that that the population mean lies in the range 25.01 and 26.19.