## 23.10 Example: Blood pressure

A US study (Schorling et al. 1997; Willems et al. 1997) examined how CHD risk factors were assessed among parts of the population with diabetes. Subjects reported to the clinic on multiple occasions. Consider this RQ:

What is the mean difference in diastolic blood pressure from the first to the second visit?

Each person has a *pair* of diastolic blood pressure (DBP) measurements:
One each from their first and second visits.
The data
(shown below)
are from the 141 people
for whom *both* measurements are available
(some data are missing).
The differences could be computed as:

- The first visit DBP minus the second visit DBP: the
*reduction*in DBP; or - The second visit DBP minus the first visit DBP: the
*increase*in DBP.

Either way is fine,
provided the order is used consistently.
Here,
the observation from the *second* visit will be used,
so that the differences
represent the *reduction* in DBP
from the first to second visit.

The parameter is \(\mu_d\),
the population mean *reduction* in DBP.

Since the data set is large,
the appropriate graphical summary is a histogram of differences
(Fig. 23.4).
The numerical summary can summarise both the first and second visit observations,
but *must* summarise the *differences*.
Numerical summaries can be computed using software,
then reported in a suitable table
(Table 23.3).

Mean | Standard deviation | Standard error | Sample size | |
---|---|---|---|---|

DBP: First visit | 94.48 | 11.473 | 0.966 | 141 |

DBP: Second visit | 92.52 | 11.555 | 0.973 | 141 |

Decrease in DBP | 1.95 | 8.026 | 0.676 | 141 |

The *standard error* of the sample mean is

\[ \text{s.e.}(\bar{d})=\frac{s_d}{\sqrt{n}} = \frac{8.02614}{\sqrt{141}} = 0.67592. \] Using an approximate multiplier of 2, the margin of error is:

\[ 2 \times 0.67592 = 1.3518, \] so an approximate 95% CI for the decrease in DBP is

\[ 1.9504\pm 1.3518, \] or from \(0.60\) to \(3.30\) mm Hg, after rounding sensibly. We write:

Based on the sample, an

approximate95% CI for the meandecreasein DBP is from \(0.60\) to \(3.30\) mm Hg.

The *exact* 95% CI from jamovi
(Fig. 23.5)
or SPSS
(Fig. 23.6),
using an exact \(t\)-multiplier rather than an approximate multiplier of 2,
is similar since the sample size is large.
After rounding,
write:

Based on the sample, an exact 95% CI for the decrease in DBP is from \(0.61\) to \(3.29\) mm Hg.

The wording (‘for the *decrease* in DBP’) implies which reading is the higher reading on average:
the first.

The CI is statistically valid as the sample size is larger than 25.
(The *data* do not need to follow a normal distribution.)

**Think 23.2 (Understanding samples)**Is there a mean difference in DBP

*in the population*?