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    5. Statistics & Probability

    Statistics & Probability

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    Question 1
    HLPaper 2

    The number of apples that Marco eats during any particular day follows a Poisson distribution with mean λ=0.2\lambda = 0.2λ=0.2.

    1.

    Find the probability that Marco eats at least one apple in a particular day.

    [2]
    2.

    Let XXX be the number of weeks in the year in which Marco eats no apples.

    Given that Marco eats apples in any given week with probability 0.8, independently of other weeks.

    Therefore, the probability of Marco not eating apples in a given week is 0.2.

    X∼B(52,0.2)X \sim B(52, 0.2)X∼B(52,0.2)

    E(X)=np=52×0.2=10.4E(X) = np = 52 \times 0.2 = 10.4E(X)=np=52×0.2=10.4

    Therefore, the expected number of weeks in which Marco eats no apples is 10.4 weeks.

    [4]
    Question 2
    SLPaper 1

    A team of 10 women recorded the number of hours they spent watching movies during a specific week. Their results are summarized in the box-and-whisker plot below.

    Box-and-Whisker Plot

    The team of women watched a total of 180 hours of movies.

    A team of 20 men also recorded the number of hours they spent watching movies that same week. Their results are summarized in the table below.

    Table

    The following week, the team of men had exams. During this exam week, the men spent half as much time watching movies compared to the previous week. For this exam week, find

    1.

    the mean number of hours that the team of men spent watching movies.

    [2]
    2.

    Find the mean number of hours that the women in this team spent watching movies that week.

    [2]
    3.

    The range of the data is 16. Find the value of a{a}a.

    [2]
    4.

    Find the value of the interquartile range.

    [2]
    5.

    Find the total number of hours the team of men spent watching movies that week.

    [2]
    6.

    Find the mean number of hours that all 30 women and men spent watching movies that week.

    [3]
    Question 3
    HLPaper 1

    A factory produces widgets using two machines, A and B. Machine A produces 60% of the widgets, and Machine B produces 40%. 2% of the widgets produced by Machine A are defective, while 5% of the widgets produced by Machine B are defective.

    1.

    Calculate the probability that a randomly selected widget is defective.

    [3]
    2.

    Given that a randomly selected widget is defective, calculate the probability that it was produced by Machine A.

    [4]
    Question 4
    SLPaper 1

    Consider a discrete random variable XXX that represents the number of heads obtained when a fair coin is tossed three times.

    1.

    Find the probability distribution of XXX.

    [4]
    2.

    Calculate the expected value of XXX.

    [3]
    Question 5
    SLPaper 1

    In a group of 30 participants, 181818 are proficient in Italian, 101010 are proficient in German, and 555 are not proficient in either of these languages. The following Venn diagram shows the events "proficient in Italian" and "proficient in German."

    The values mmm, nnn, ppp, and qqq represent numbers of participants.

    Venn Diagram

    1.

    Write down the value of qqq.

    [1]
    2.

    Write down the value of mmm and of ppp.

    [3]
    3.

    Find the value of n{n}n.

    [2]
    Question 6
    SLPaper 1

    Consider a data set representing the scores of 50 students in a mathematics test.

    1.

    Calculate the mean score of the students if the scores are: 45, 50, 55, 60, 65, 70, 75, 80, 85, 90.

    [2]
    2.

    Determine the median score of the students.

    [2]
    3.

    Identify the mode of the scores.

    [1]
    4.

    Calculate the standard deviation of the scores.

    [3]
    Question 7
    SL & HLPaper 1

    Consider a normal distribution with mean μ=50\mu = 50μ=50 and standard deviation σ=10\sigma = 10σ=10.

    1.

    Find the probability that a randomly selected value from this distribution is less than 40.

    [3]
    2.

    Find the probability that a randomly selected value from this distribution is between 45 and 55.

    [3]
    3.

    Determine the value of xxx such that 90% of the distribution is below xxx.

    [4]
    Question 8
    HLPaper 1

    Let YYY be a continuous random variable with probability density function f(y)={38y2,0≤y≤20,otherwisef(y) = \begin{cases} \frac{3}{8}y^2, & 0 \leq y \leq 2 \\ 0, & \text{otherwise} \end{cases}f(y)={83​y2,0,​0≤y≤2otherwise​.

    1.

    Verify that f(y)f(y)f(y) is a valid probability density function.

    [3]
    2.

    Find the expected value of YYY.

    [4]
    Question 9
    SL & HLPaper 1

    A factory produces light bulbs with lifetimes that are normally distributed with a mean of 800 hours and a standard deviation of 100 hours.

    1.

    Calculate the probability that a randomly selected light bulb lasts more than 950 hours.

    [3]
    2.

    Find the probability that a randomly selected light bulb lasts between 700 and 900 hours.

    [3]
    3.

    Determine the lifetime that only 5% of the light bulbs exceed.

    [4]
    Question 10
    HLPaper 1

    A medical test for a certain disease has a 98% sensitivity and a 95% specificity. The prevalence of the disease in the population is 1%.

    1.

    Calculate the probability that a person selected at random from the population tests positive for the disease.

    [3]
    2.

    Given that a person tests positive, calculate the probability that they actually have the disease.

    [4]
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