Question

7.46 Young Adults at Risk. Research by R. Pyhala et al. shows that young adults who were born prematurely with very low birth weights (below $1500$grams) have higher blood pressure than those born at term. The study can be found in the article. "Blood Pressure Responses to Physiological Stress in Young Adults with Very Low Birth Weight" (Pediatrics, Vol. 123, No, 2, pp. $731-734$). The researchers found that systolic blood pressures, of young adults who were born prematurely with very low birth weights have mean $120.7\mathrm{mmHg}$and standard deviation $13.8\mathrm{mmHg}$.
a. Identify the population and variable.
b. For samples of $30$ young adults who were born prematurely with very low birth weights, find the mean and standard deviation of all possible sample mean systolic blood pressures. Interpret your results in words.
c. Repeat part (b) for samples of size $90$ .

Short answer

(a) The population is comprised of all young individuals who were born prematurely and with extremely low birth weights, and the variable is systolic blood pressure.

(b) The mean and standard deviation of all possible sample systolic blood pressures are$120.7\mathrm{mmHg}$and $2.52\mathrm{mmHg}$.

(c) The mean and standard deviation of all possible sample systolic blood pressures are $120.7\mathrm{mmHg}$ and $1.45mmHg$.

Step-by-step solution

Part (a) Step 1: Given information

To identify the population and variable. Note that the researchers found that systolic blood pressures, of young adults who were born prematurely with very low birth weights have mean $120.7mmHg$and standard deviation $13.8mmHg$.

Part (a) Step 2: Explanation

The population is identified by:

The population is made up of all of the people who are being studied.
In other terms, the population is defined as the collection of all individuals, items, or objects required for a certain study.
The participants in this study are young adults who were born prematurely and with extremely low birth weights.
As a result, the population consists entirely of young adults who were born prematurely and at extremely low birth weights.
The variable is identified by:
An property or a quality that may be measured is referred to as a variable.
Each unit's value for the variable may be different.
In other words, a variable is a trait that is recorded for each case.
The systolic pressure of young individuals was assessed in this investigation.
Systolic blood pressure clearly differs from person to person.
As a result, systolic blood pressure is the variable.

Therefore, the population is comprised of all young individuals who were born prematurely and with extremely low birth weights, and the variable is systolic blood pressure.

Part (b) Step 1: Given information

To determine the mean and standard deviation of all possible samples mean systolic blood pressures.

Part (b) Step 2: Explanation

The mean of all possible sample mean systolic blood pressures of sample size $30$is $\left({\mu }_x\right)$.
The mean $\left(\mu \right)$for all young adults born preterm with very low birth weights is estimated to be$120.7mmHg$
${\mu }_x=\mu$
$=120.7$
As a result, the average of all conceivable mean systolic blood pressures of sample size$30$is $120.7mmHg$.

Part (b) Step 3: Explanation

The standard deviation $\left({\sigma }_X\right)$of all possible sample mean systolic blood pressures of sample size $n$is $30$. And the standard deviation $\mu$for all young adults who were born preterm with extremely low birth weights is $13.8mmHg$.

$\begin{array}{r}{\sigma }_X=\frac{\sigma }{\sqrt{n}}\end{array}$

$=\frac{13.8}{\sqrt{30}}$

$=\frac{13.8}{5.4772}$

$\begin{array}{r}=2.52\end{array}$

Therefore, all possible sample mean systolic blood pressures of sample size have a standard deviation of $2.52mmHg$.

The mean and standard deviation of all possible sample systolic blood pressures for size 30 young people who were born preterm with very low birth weights are $120.7mmHg$and $2.52mmHg$, respectively.

Part (c) Step 1: Given information

To find the repeat part(b) for sample size $90$.

Part (c) Step 2: Explanation

The mean of all possible sample mean systolic blood pressures of sample size $90$is $\left({\mu }_x\right)$ as:
${\mu }_x=\mu$
$=120.7$
As a result, the mean $\left({\mu }_x\right)$of all sample mean systolic blood pressures of sample size $90$is $120.7mmHg$.
The standard deviation $\left(\sigma \right)$for all young people born preterm with very low birth weights is given to be $13.8mmHg$.
${\sigma }_X=\frac{\sigma }{\sqrt{n}}$

$=\frac{13.8}{\sqrt{90}}$

$=1.45$

Hence, the standard deviation $\left({\sigma }_X\right)$ of all possible sample mean systolic blood pressures of sample size is $1.45mmHg$.