Page 1 of 4 worksheet on making inferences from random samples, featuring two problems involving statistical estimation based on survey results.

Problem 1: Making inferences from random samples (Museum Visitors)

#### Question:
A museum employee surveyed a random sample of 40 visitors to the museum about their purchases at the museum. Of the visitors surveyed:
- 11 bought something at the gift shop,
- 13 went to the coffee shop, and
- 16 made a purchase from the souvenir shop.

Based on these results, how many people out of the next 100 visitors to the museum would be expected to make a purchase from the souvenir shop?

#### Solution:
To solve this problem, we need to use the proportion of visitors who made a purchase from the souvenir shop in the sample and apply it to the next 100 visitors.

1. Calculate the proportion of visitors who made a purchase from the souvenir shop:
- Number of visitors who made a purchase from the souvenir shop = 16
- Total number of visitors surveyed = 40
- Proportion = \(\frac{16}{40} = 0.4\) or 40%

2. Apply this proportion to the next 100 visitors:
- Expected number of visitors who will make a purchase from the souvenir shop = \(0.4 \times 100 = 40\)

#### Final Answer for Problem 1:
\[
\boxed{40}
\]

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Problem 2: Making inferences from random samples (Favorite Season)

#### Question:
Francis surveyed a random sample of 170 students at Franklin High School about their favorite season. Of the students surveyed:
- 38 chose fall as their favorite season,
- 28 chose summer,
- 46 chose spring,
- 52 chose winter, and
- 6 had no preference.

There are 1816 students at Franklin High School. Based on Francis's survey data, predict the number of students at Franklin High School whose favorite season is fall.

#### Solution:
To solve this problem, we need to use the proportion of students who chose fall as their favorite season in the sample and apply it to the total student population.

1. Calculate the proportion of students who chose fall as their favorite season:
- Number of students who chose fall = 38
- Total number of students surveyed = 170
- Proportion = \(\frac{38}{170}\)

2. Simplify the proportion:
- \(\frac{38}{170} = \frac{19}{85}\) (simplified fraction)

3. Apply this proportion to the total student population:
- Total number of students at Franklin High School = 1816
- Expected number of students who favor fall = \(\frac{19}{85} \times 1816\)

4. Perform the calculation:
\[
\frac{19}{85} \times 1816 = \frac{19 \times 1816}{85} = \frac{34504}{85} \approx 405.93
\]
Since the number of students must be a whole number, we round to the nearest whole number:
\[
405.93 \approx 406
\]

#### Final Answer for Problem 2:
\[
\boxed{406}
\]

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Summary of Answers:

1. Problem 1: \(\boxed{40}\)
2. Problem 2: \(\boxed{406}\)