抽樣、資料類型與偏差：練習題

A survey was conducted among $$50$$ students about their music preferences. It was found that $$30$$ students enjoy classical music, $$15$$ students enjoy rock music, and $$5$$ students enjoy both classical and rock music. If a student is chosen randomly from this group, what is the probability that the student enjoys classical music only?

A certain type of electronic component has a probability of failure of \(0.05\). A batch of \(100\) such components is randomly selected for testing.

Assuming the failures are independent, what is the standard deviation of the number of failed components in this batch?

The lifetimes of a certain brand of light bulbs are normally distributed. It is known that \(10\%\) of the bulbs last less than 800 hours, and \(5\%\) of the bulbs last more than 1200 hours. Estimate the mean lifetime of these light bulbs, to the nearest integer hour.

Use Z-values: P(Z < -1.2816) = 0.10 and P(Z < 1.6449) = 0.95.

Events \(A\) and \(B\) are independent. Given that \(P(A) = 0.4\) and \(P(B) = 0.5\), find \(P(A \cup B)\).

A survey was conducted among \(100\) students regarding their gender and whether they passed a recent exam. The results are summarized in the table below:

If a randomly selected student is male, what is the probability that this student passed the exam?

The heights of a certain population are normally distributed with a mean of \(170\) cm and a standard deviation of \(5\) cm. What is the probability that a randomly selected person has a height between \(165\) cm and \(175\) cm? Give your answer to four decimal places.

The following data shows the preferred sport of 100 students, classified by gender:
Male: Basketball (15), Football (35)
Female: Basketball (25), Football (25)

Calculate the expected frequency of males who prefer football, assuming that the preferred sport is independent of gender.

A factory produces light bulbs, and it is known that $$5\%$$ of the bulbs are defective. A quality control test correctly identifies a defective bulb $$90\%$$ of the time (true positive rate) and incorrectly flags a non-defective bulb $$2\%$$ of the time (false positive rate). If a randomly selected bulb is flagged as defective by the test, what is the probability that it is actually defective?

A group of 8 students recorded the number of hours they spent studying for a mathematics test: 5, 8, 3, 5, 7, 6, 5, 9.

Calculate:

a) The mean number of hours studied.

b) The median number of hours studied.

c) The mode number of hours studied.

The weights of adult male lions in a wildlife reserve are normally distributed with a mean of \(190\) kg and a standard deviation of \(15\) kg.

a) A male lion is randomly selected. Find the probability that its weight is less than \(170\) kg. Give your answer to four decimal places.

b) The reserve wants to identify the heaviest \(10\%\) of male lions. Find the minimum weight, in kg, that a male lion must have to be considered in this heaviest \(10\%\). Give your answer to one decimal place.