The data set lowbwt contains information for a sample of 100 low birth weight infants born in two teaching hospitals in Boston, Massachusetts. Measurements of systolic blood pressure are saved under the variable name sbp and indicators of gender-with 1 representing a male and 0 a female-under the name sex. sbp sex tox grmhem gestage apgar5 43 1 0 0 29 7 51 1 0 0 31 8 42 0 0 0 33 0 39 0 0 0 31 8 48 0 1 0 30 7 31 1 0 1 25 0 31 1 1 0 27 7 40 0 0 0 29 9 57 0 0 0 28 6 64 0 1 0 29 9 46 0 0 0 26 7 47 0 0 1 30 6 63 0 0 0 29 8 56 0 0 0 29 1 49 1 0 0 29 8 87 1 0 0 29 7 46 0 0 0 29 8 66 0 0 0 33 8 42 0 1 0 33 8 52 0 0 0 29 7 51 1 0 0 28 7 47 0 0 0 30 9 54 1 0 0 27 4 64 1 0 0 33 9 37 0 0 0 32 7 36 1 1 0 28 3 45 0 0 1 29 7 39 1 0 0 28 7 29 0 0 0 29 4 61 0 0 0 30 3 53 1 0 0 31 7 64 0 0 0 30 7 35 0 0 0 31 6 34 1 0 0 29 9 62 0 0 0 27 7 59 0 0 0 27 8 36 1 0 0 27 9 47 0 0 0 32 8 45 1 0 1 31 2 62 0 0 1 28 5 75 1 1 0 30 7 44 1 0 0 29 0 39 1 0 0 28 8 48 0 0 1 31 7 43 0 1 0 27 6 19 0 0 1 25 4 63 1 0 0 30 7 42 1 0 0 28 6 44 0 0 0 28 9 25 0 0 0 25 8 26 0 0 0 23 8 27 1 0 0 27 9 35 1 0 0 28 8 40 1 0 0 27 7 44 0 0 0 27 6 66 1 0 0 26 8 59 0 0 0 25 3 24 0 0 0 23 7 40 0 0 1 26 3 49 0 0 0 24 5 53 1 1 0 29 9 45 0 0 0 29 9 50 1 0 1 27 8 64 1 0 0 30 7 48 0 0 0 30 6 48 0 0 1 32 4 58 0 1 0 33 7 67 0 0 0 27 8 40 0 0 1 31 8 48 0 0 0 26 8 36 1 0 0 27 5 44 1 0 0 27 6 53 0 1 0 35 9 45 0 1 0 28 6 54 1 0 0 30 8 44 1 1 0 31 2 42 1 0 0 30 5 50 0 0 0 27 0 48 0 0 0 25 5 29 0 0 1 25 5 30 0 0 1 26 2 36 0 0 0 29 0 44 0 0 0 29 0 46 0 1 0 34 9 51 1 1 0 30 4 51 1 0 0 29 5 43 1 1 0 33 7 48 1 0 0 30 5 52 1 0 0 29 8 43 1 0 0 24 6 42 1 1 0 33 8 48 1 0 1 25 5 49 0 1 0 32 8 62 1 1 0 31 7 45 1 0 0 31 9 51 0 1 1 31 6 52 1 0 0 29 8 47 1 1 0 32 5 40 0 1 0 33 8 50 0 0 0 28 7 (a). Using systolic blood pressure as the response and gestational age as the explanatory variable, compute the least-squares regression line. Interpret the estimated slope and y-intercept of the line; what do they mean in words? (b). At the 0.05 level of significance, test the null hypothesis that the true population slope β is equal to 0. What do you conclude? (c). What is the estimated mean systolic blood pressure for the population of low birth weight infants whose gestational age is 31 weeks?
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
The data set lowbwt contains information for a sample of 100 low birth weight infants born in two teaching hospitals in Boston, Massachusetts. Measurements of systolic blood pressure are saved under the variable name sbp and indicators of gender-with 1 representing a male and 0 a female-under the name sex.
|
sbp |
sex | tox | grmhem | gestage | apgar5 |
| 43 | 1 | 0 | 0 | 29 | 7 |
| 51 | 1 | 0 | 0 | 31 | 8 |
| 42 | 0 | 0 | 0 | 33 | 0 |
| 39 | 0 | 0 | 0 | 31 | 8 |
| 48 | 0 | 1 | 0 | 30 | 7 |
| 31 | 1 | 0 | 1 | 25 | 0 |
| 31 | 1 | 1 | 0 | 27 | 7 |
| 40 | 0 | 0 | 0 | 29 | 9 |
| 57 | 0 | 0 | 0 | 28 | 6 |
| 64 | 0 | 1 | 0 | 29 | 9 |
| 46 | 0 | 0 | 0 | 26 | 7 |
| 47 | 0 | 0 | 1 | 30 | 6 |
| 63 | 0 | 0 | 0 | 29 | 8 |
| 56 | 0 | 0 | 0 | 29 | 1 |
| 49 | 1 | 0 | 0 | 29 | 8 |
| 87 | 1 | 0 | 0 | 29 | 7 |
| 46 | 0 | 0 | 0 | 29 | 8 |
| 66 | 0 | 0 | 0 | 33 | 8 |
| 42 | 0 | 1 | 0 | 33 | 8 |
| 52 | 0 | 0 | 0 | 29 | 7 |
| 51 | 1 | 0 | 0 | 28 | 7 |
| 47 | 0 | 0 | 0 | 30 | 9 |
| 54 | 1 | 0 | 0 | 27 | 4 |
| 64 | 1 | 0 | 0 | 33 | 9 |
| 37 | 0 | 0 | 0 | 32 | 7 |
| 36 | 1 | 1 | 0 | 28 | 3 |
| 45 | 0 | 0 | 1 | 29 | 7 |
| 39 | 1 | 0 | 0 | 28 | 7 |
| 29 | 0 | 0 | 0 | 29 | 4 |
| 61 | 0 | 0 | 0 | 30 | 3 |
| 53 | 1 | 0 | 0 | 31 | 7 |
| 64 | 0 | 0 | 0 | 30 | 7 |
| 35 | 0 | 0 | 0 | 31 | 6 |
| 34 | 1 | 0 | 0 | 29 | 9 |
| 62 | 0 | 0 | 0 | 27 | 7 |
| 59 | 0 | 0 | 0 | 27 | 8 |
| 36 | 1 | 0 | 0 | 27 | 9 |
| 47 | 0 | 0 | 0 | 32 | 8 |
| 45 | 1 | 0 | 1 | 31 | 2 |
| 62 | 0 | 0 | 1 | 28 | 5 |
| 75 | 1 | 1 | 0 | 30 | 7 |
| 44 | 1 | 0 | 0 | 29 | 0 |
| 39 | 1 | 0 | 0 | 28 | 8 |
| 48 | 0 | 0 | 1 | 31 | 7 |
| 43 | 0 | 1 | 0 | 27 | 6 |
| 19 | 0 | 0 | 1 | 25 | 4 |
| 63 | 1 | 0 | 0 | 30 | 7 |
| 42 | 1 | 0 | 0 | 28 | 6 |
| 44 | 0 | 0 | 0 | 28 | 9 |
| 25 | 0 | 0 | 0 | 25 | 8 |
| 26 | 0 | 0 | 0 | 23 | 8 |
| 27 | 1 | 0 | 0 | 27 | 9 |
| 35 | 1 | 0 | 0 | 28 | 8 |
| 40 | 1 | 0 | 0 | 27 | 7 |
| 44 | 0 | 0 | 0 | 27 | 6 |
| 66 | 1 | 0 | 0 | 26 | 8 |
| 59 | 0 | 0 | 0 | 25 | 3 |
| 24 | 0 | 0 | 0 | 23 | 7 |
| 40 | 0 | 0 | 1 | 26 | 3 |
| 49 | 0 | 0 | 0 | 24 | 5 |
| 53 | 1 | 1 | 0 | 29 | 9 |
| 45 | 0 | 0 | 0 | 29 | 9 |
| 50 | 1 | 0 | 1 | 27 | 8 |
| 64 | 1 | 0 | 0 | 30 | 7 |
| 48 | 0 | 0 | 0 | 30 | 6 |
| 48 | 0 | 0 | 1 | 32 | 4 |
| 58 | 0 | 1 | 0 | 33 | 7 |
| 67 | 0 | 0 | 0 | 27 | 8 |
| 40 | 0 | 0 | 1 | 31 | 8 |
| 48 | 0 | 0 | 0 | 26 | 8 |
| 36 | 1 | 0 | 0 | 27 | 5 |
| 44 | 1 | 0 | 0 | 27 | 6 |
| 53 | 0 | 1 | 0 | 35 | 9 |
| 45 | 0 | 1 | 0 | 28 | 6 |
| 54 | 1 | 0 | 0 | 30 | 8 |
| 44 | 1 | 1 | 0 | 31 | 2 |
| 42 | 1 | 0 | 0 | 30 | 5 |
| 50 | 0 | 0 | 0 | 27 | 0 |
| 48 | 0 | 0 | 0 | 25 | 5 |
| 29 | 0 | 0 | 1 | 25 | 5 |
| 30 | 0 | 0 | 1 | 26 | 2 |
| 36 | 0 | 0 | 0 | 29 | 0 |
| 44 | 0 | 0 | 0 | 29 | 0 |
| 46 | 0 | 1 | 0 | 34 | 9 |
| 51 | 1 | 1 | 0 | 30 | 4 |
| 51 | 1 | 0 | 0 | 29 | 5 |
| 43 | 1 | 1 | 0 | 33 | 7 |
| 48 | 1 | 0 | 0 | 30 | 5 |
| 52 | 1 | 0 | 0 | 29 | 8 |
| 43 | 1 | 0 | 0 | 24 | 6 |
| 42 | 1 | 1 | 0 | 33 | 8 |
| 48 | 1 | 0 | 1 | 25 | 5 |
| 49 | 0 | 1 | 0 | 32 | 8 |
| 62 | 1 | 1 | 0 | 31 | 7 |
| 45 | 1 | 0 | 0 | 31 | 9 |
| 51 | 0 | 1 | 1 | 31 | 6 |
| 52 | 1 | 0 | 0 | 29 | 8 |
| 47 | 1 | 1 | 0 | 32 | 5 |
| 40 | 0 | 1 | 0 | 33 | 8 |
| 50 | 0 | 0 | 0 | 28 | 7 |
(a). Using systolic blood pressure as the response and gestational age as the explanatory variable, compute the least-squares regression line. Interpret the estimated slope and y-intercept of the line; what do they mean in words?
(b). At the 0.05 level of significance, test the null hypothesis that the true population slope β is equal to 0. What do you conclude?
(c). What is the estimated mean systolic blood pressure for the population of low birth weight infants whose gestational age is 31 weeks?
(d). Construct 95% confidence interval for the true mean value of systolic blood pressure when x=31 weeks.
(e). Suppose that you randomly select a new child from the population of low birth weight infants with gestational age 31 weeks. What is the predicted systolic blood pressure for this child?
(f). Construct a 95% prediction interval for this new value of systolic blood pressure.
(g). Does the lease-squares regression model seem to fit the observed data? Comment on the coefficient of determination and a plot of the residuals versus the fitted values of systolic blood pressure.
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