Question

data from four identical trials on a new sleep herb are shown in the table. complete parts (a) and (b).

group	number of subjects	number who reported better sleep
b	170	30
c	210	30
d	180	40

(round to the nearest integer as needed.)
a. the herb is effective in improving sleep for about 22 % of all people.
b. in any group of people, the herb is effective in improving sleep for about \\(\square\\) people.
b. the next trial will have 1,000 subjects. what is a reasonable prediction for the next trial?
in the next trial, the herb should be effective in improving sleep for about \\(\square\\) people.
(round to the nearest integer as needed.)

Explanation:

To solve part (b), we first need to find the overall proportion of people who reported better sleep across all groups, and then use that proportion to predict the number of people in a trial with 1,000 subjects.

Step 1: Calculate the total number of subjects

We sum the number of subjects in each group:
\[
200 + 170 + 210 + 180 = 760
\]

Step 2: Calculate the total number of people who reported better sleep

We sum the number of people who reported better sleep in each group:
\[
50 + 30 + 40 + 40 = 160
\]

Step 3: Find the proportion of people who reported better sleep

The proportion \( p \) is the total number of people who reported better sleep divided by the total number of subjects:
\[
p = \frac{160}{760} \approx 0.2105
\]

Step 4: Predict the number of people in a trial with 1,000 subjects

We multiply the proportion by 1,000:
\[
1000 \times 0.2105 \approx 211
\]

Answer:

In the next trial, the herb should be effective in improving sleep for about \(\boxed{211}\) people.