Matemática Elementar

Problema 1

A school has $800$ students. A researcher wants to estimate how many students prefer pizza for lunch. She randomly surveys $50$ students and finds that $32$ prefer pizza. Estimate the number of students in the entire school who prefer pizza.

Problemas

1. A school has $800$ students. A researcher wants to estimate how many students prefer pizza for lunch. She randomly surveys $50$ students and finds that $32$ prefer pizza. Estimate the number of students in the entire school who prefer pizza.

2. Explain why each of the following sampling methods might produce biased results:

a) Surveying only students in the cafeteria to find out what all students think about school lunch.

b) Asking every 10th person entering a mall on a Saturday about their favorite weekday activity.

3. A bag contains $5$ red marbles, $3$ blue marbles, and $2$ green marbles. You draw one marble at random. Find the probability of each event and classify it as impossible, unlikely, equally likely as not, likely, or certain.

a) Drawing a red marble.

b) Drawing a yellow marble.

c) Drawing a marble that is red or blue.

4. A student flips a coin $200$ times and gets heads $114$ times. What is the experimental probability of getting heads? How does it compare to the theoretical probability?

5. A factory produces light bulbs. In a random sample of $250$ bulbs, $8$ were defective. The factory produces $10{,}000$ bulbs per day.

a) What is the experimental probability that a bulb is defective?

b) Based on this sample, how many defective bulbs would you predict are produced per day?

c) Is this estimate guaranteed to be exact? Explain.

Gabarito — Statistics and Probability

1. In the sample, the fraction who prefer pizza is:

$\frac{32}{50} = 0.64 = 64\%$

To estimate the total in the school:

$0.64 \times 800 = 512$

Approximately $512$ students in the school are estimated to prefer pizza.

2. a) Bias: Only students in the cafeteria are surveyed. Students who bring lunch from home or eat elsewhere are excluded. Those in the cafeteria are more likely to have opinions about (or prefer) the cafeteria food, so the sample does not represent all students.

b) Bias: The survey is conducted on a Saturday at a mall. People at a mall on Saturday are likely to enjoy shopping and leisure activities, and may not represent the general population's weekday activity preferences. Also, people who work on Saturdays or don't go to malls are excluded.

3. Total marbles: $5 + 3 + 2 = 10$ .

a) $P(\text{red}) = \frac{5}{10} = \frac{1}{2} = 0.5$ . This is equally likely as not (exactly $50\%$ ).

b) $P(\text{yellow}) = \frac{0}{10} = 0$ . This is impossible — there are no yellow marbles.

c) $P(\text{red or blue}) = \frac{5 + 3}{10} = \frac{8}{10} = \frac{4}{5} = 0.8$ . This is likely (probability greater than $0.5$ but less than $1$ ).

4. The experimental probability is:

$P(\text{heads}) = \frac{114}{200} = 0.57 = 57\%$

The theoretical probability of heads for a fair coin is:

$P(\text{heads}) = \frac{1}{2} = 0.50 = 50\%$

The experimental probability ( $57\%$ ) is slightly higher than the theoretical probability ( $50\%$ ). This difference is likely due to random variation. As the number of flips increases, the experimental probability should get closer to $50\%$ (this is the law of large numbers).

5. a) The experimental probability of a defective bulb:

$P(\text{defective}) = \frac{8}{250} = 0.032 = 3.2\%$

b) Predicted defective bulbs per day:

$0.032 \times 10{,}000 = 320 \text{ bulbs}$

c) No, this estimate is not guaranteed to be exact. It is a prediction based on one sample. Different random samples could give slightly different defect rates. The prediction assumes the sample is representative of the full production. Larger samples and repeated sampling would give a more reliable estimate.