The United States Census Bureau has calculated the average age of marriage every 10 vears for the past 120 years. When calculating the regression line for Years Since 1900 vs Average Age of Marriage they found the model to be \[ \text { Average Age of Marriage }=20.4+0.08 \text { (Years Since 1900). } \] i) If the standard deviation for Years is 30 and the standard deviation for Average Age is 4.23, determine the correlation coefficient between Years and Average Age. ii) Find \( R^{2} \). Explain what this means in the context of this problem. iii) Predict the average age of marriage in the year 1984 based on the model above. iv) It turns out the actual average age of marriage in 1984 was actually 24 . What is the residual for this year? Did our model overestimate or underestimate? v) Would we want to use this model to predict the average age of marriage in the year 2200 ? Why or why not?
Added by Grace S.
Close
Step 1
In the context of a simple linear regression model: \[ r = \frac{\beta s_x}{s_y} \] where \( \beta \) is the slope of the regression line. Plugging in the values: \[ \beta = 0.08, \quad s_x = 30, \quad s_y = 4.23 \] \[ r = \frac{0.08 \times 30}{4.23} \approx Show more…
Show all steps
Your feedback will help us improve your experience
Jainendra Ojha and 95 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Marriage In 1960, census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has increased during the past 40 years. a) Write appropriate hypotheses. b) We plan to test our hypothesis by selecting a random sample of 40 men who married for the first time last year. Do you think the necessary assumptions for inference are satisfied? Explain. c) Describe the approximate sampling distribution model for the mean age in such samples. d) The men in our sample married at an average age of 24.2 years, with a standard deviation of 5.3 years. What’s the P-value for this result? e) Explain (in context) what this P-value means. f) What’s your conclusion?
Learning About the World
Inferences About Means
Marriage The following table lists the U.S. median age at first marriage for men and women. The age at which both groups marry for the first time seems to be increasing at a roughly linear rate in recent decades. Let $t$ correspond to the number of years since $1980 .$ Source: U.S. Census Bureau. (a) Find a linear equation that approximates the data for men, using the data for the years 1980 and $2010 .$ (b) Repeat part (a) using the data for women. (c) Which group seems to have the faster increase in median age at first marriage? (d) According to the equation from part (a), in what year will the men's median age at first marriage reach 30$?$ (e) When the men's median age at first marriage is $30,$ what will the median age be for women?
Linear Functions
Slopes and Equations of Lines
In 1980, census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has increased during the past 40 years. A random sample of 40 men who married for the first time last year is selected, and the results found a mean of 24.2 years and a standard deviation of 5.3 years? Is there evidence the mean age at first marriage for men has increased. Test using a 5% significance level.
Madhur L.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD