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 the x - axis from 35.5 to 38.5 and dots representing data points
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. michaels temperature was 37.5°c. calculate and interpret michaels 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:

Step1: Compute position of 10th percentile

Position = $\frac{10}{100} \times 130 = 13$

Step2: Identify 13th value (10th percentile)

From data, the 13th value is ~36.4°C (or using linear approx: Min=35.722, Q1=36.556; 13th value is near lower end, approx 36.4°C)

Step3: Calculate z-score for 37.5°C

$z = \frac{x - \text{Mean}}{\text{SD}} = \frac{37.5 - 36.805}{0.407} \approx 1.71$

Step4: Check normality (dotplot + empirical rule)

Dotplot is roughly symmetric; Empirical rule: Mean±1SD=36.805±0.407=(36.398,37.212) (covers ~68%? Data shows most values here); Mean±2SD=(35.991,37.619) (covers ~95%); Mean±3SD=(35.584,38.026) (covers almost all except max=38.222). So approx normal.

Step5: Convert mean to Fahrenheit

Mean°F = $\frac{9}{5} \times 36.805 + 32 = 98.249°F$

Step6: Convert SD to Fahrenheit

SD°F = $\frac{9}{5} \times 0.407 = 0.7326°F$

Answer:

9a. ~36.4°C
9b. z≈1.71 (37.5°C is ~1.71 SD above mean)
10a. Yes (symmetric dotplot; empirical rule holds)
10b. Mean°F≈98.25, SD°F≈0.733