Question

section ii: free response
questions 9 and 10 refer to this setting. for more than a century, doctors have been telling patients that a normal body temperature is 98.6°f (37.0°c). this value dates back to a study done by carl sunderlich in the mid-1800s. more recently, researchers conducted a study to determine whether the \accepted\ value for normal body temperature is accurate. they collected body temperatures (in degrees celsius) from 130 healthy individuals. a dotplot and numerical summaries of the data are given.
dotplot image with temperature (°c) on x - axis from 35.5 to 38.5
table with columns n, mean, sd, min, q₁, med, q₃, max and row 130, 36.805, 0.407, 35.722, 36.556, 36.833, 37.056, 38.222

1. a. find the 10th percentile of the distribution of body temperature.

b. michael’s temperature was 37.5°c. calculate and interpret michael’s standardized score (z - score).

1. a. is this distribution of body temperature approximately normal? justify your answer based on the graph and the empirical rule.

b. suppose that the body temperatures are converted from celsius to fahrenheit using the formula (9/5)c° + 32 = f°. find the mean and standard deviation of the transformed values.

Explanation:

9a Step1: Calculate position of 10th percentile

Position = $0.10 \times n = 0.10 \times 130 = 13$

9a Step2: Identify 13th data point

From ordered data, 13th value is ~36.4°C (or using linear approx if needed)

9b Step1: Apply z-score formula

$z = \frac{x - \mu}{\sigma} = \frac{37.5 - 36.805}{0.407}$

9b Step2: Compute z-score

$z \approx \frac{0.695}{0.407} \approx 1.71$

10a Step1: Check dotplot shape

Dotplot is roughly symmetric and bell-shaped

10a Step2: Apply empirical rule

Mean ± 1SD: $36.805 ± 0.407 = (36.398, 37.212)$ (most data here); Mean ± 2SD: $(35.991, 37.619)$ (almost all data) – consistent with normal

10b Step1: Convert mean to Fahrenheit

$\text{Mean}_F = \frac{9}{5} \times 36.805 + 32 = 66.249 + 32 = 98.249$

10b Step2: Convert SD to Fahrenheit

$\text{SD}_F = \frac{9}{5} \times 0.407 = 0.7326$

Answer:

9a. ~36.4°C
9b. z ≈ 1.71; Michael’s temperature is ~1.71 SD above the mean
10a. Yes; symmetric dotplot and empirical rule alignment
10b. Mean: 98.25°F, SD: 0.73°F