You are interested if there is a relationship between the average monthly temperature 6 months after birth (x) and the average age in weeks that the child crawled (y).
| birth month | avg_crawling_age in weeks(y) | Temperature (x) |
|---|---|---|
| January | 29.84 | 66 |
| February | 30.52 | 73 |
| March | 29.7 | 72 |
| April | 31.84 | 63 |
| May | 28.58 | 52 |
| June | 31.44 | 39 |
| July | 33.64 | 33 |
| August | 32.82 | 30 |
| September | 33.83 | 33 |
| October | 33.35 | 37 |
| November | 33.38 | 48 |
| December | 32.32 | 57 |
a. Given the data above, what is the correlation coefficient?
b. What is the p value obtained when testing if the correlation coefficient?
c. Is there a linear relationship between temperature 6 months after birth and the average crawling age of babies at .05 significance level?
d. Assuming that there is a linear relationship, what is the line that describes the relationship?
e. If my daughter was born in a month where the average temperature for the 6 months after she was born was 36 degrees, what age would you expect her to crawl at?
Answer (1)
Calculate the correlation coefficient between temperature and crawling age: r ≈ − 0.6997 .
Determine the p-value for the correlation coefficient: p ≈ 0.0113 .
Test for a linear relationship at a 0.05 significance level: Since p < 0.05 , there is a linear relationship.
Find the regression line: y = 35.6781 − 0.0777 x , and predict crawling age for x = 36: y ≈ 32.8794 .
Explanation
Problem Analysis We are given a dataset of birth months, average crawling age in weeks (y), and average temperature (x) 6 months after birth. We need to find the correlation coefficient, p-value, determine if there is a linear relationship, find the regression line, and predict the crawling age for a given temperature.
Calculating Correlation Coefficient First, we calculate the correlation coefficient (r) between x and y. The correlation coefficient measures the strength and direction of a linear relationship between two variables. The result is approximately -0.6997.
Calculating P-value Next, we calculate the p-value to determine the statistical significance of the correlation coefficient. The p-value is the probability of observing a correlation coefficient as extreme as, or more extreme than, the one calculated from the sample data, assuming that there is no actual relationship between the variables. The calculated p-value is approximately 0.0113.
Testing for Linear Relationship We compare the p-value to the significance level of 0.05. Since 0.0113 < 0.05, we reject the null hypothesis and conclude that there is a statistically significant linear relationship between temperature 6 months after birth and the average crawling age of babies.
Finding the Regression Line Now, we determine the equation of the line that describes the relationship. This is the regression line, which is given by y = a + b x , where a is the y-intercept and b is the slope. The calculated slope (b) is approximately -0.0777, and the y-intercept (a) is approximately 35.6781. Therefore, the regression line is y = 35.6781 − 0.0777 x .
Predicting Crawling Age Finally, we predict the crawling age (y) for a baby born in a month where the average temperature for the 6 months after she was born was 36 degrees. We plug x = 36 into the regression equation: y = 35.6781 − 0.0777 ( 36 ) = 32.8794 .
Examples
Understanding the relationship between environmental factors like temperature and developmental milestones such as crawling age can help pediatricians and parents anticipate developmental patterns. For instance, if a region experiences consistently high temperatures during certain months, this model could help predict whether babies born during those months might crawl slightly earlier or later. This information can be valuable for early intervention and support, ensuring that children receive appropriate care and attention during their developmental stages. By analyzing such correlations, healthcare providers can offer more tailored advice and resources to families, promoting optimal child development.
