Exercise 8 Page 719

Practice makes perfect

a A survey asked female and male students about whether they prefer to take gym class or choir. The findings are presented in the following two-way table, which we are asked to complete.

Let's start with the bottom row since it contains the most information. We see that a total of 106 students were asked about their preferences and that 49 of them chose choir. This gives us the number of students who preferred gym class.

Next, if we look at the first column, we see that a total of 57 students chose gym class, out of whom 23 were female. By subtracting the two numbers we get the number of males who chose gym class.

Following similar steps, we can find all the other frequencies in the table. We need to remember to always move to a column or a row where only one value is missing.

b We want to find the probability that a randomly selected student is female and prefers choir. To do so, let's take a look at the completed table from Part A.

We see that there are 33 females who prefer choir to gym. Now, the probability that a randomly selected student is female and prefers choir is the relative frequency of such students. Female Students Who Prefer Choir/Total In the table we see that there are a total of 106 surveyed students. Let's substitute the numbers and calculate the quotient. 33/106 &= 0.311320... & ≈ 0.311 The probability that a randomly selected student is female and prefers choir is approximately 0.311, or about 31.1 %.

c We want to know the probability that a randomly selected male student prefers gym class. We will start with finding the joint relative frequencies for our table. To do that, we need to divide each frequency by the total number of students in the survey, 106.

Each of those frequencies represents some probability. For example, there is a 0.151=15.1 % probability that a surveyed student is male and prefers choir. Moving on, we can find the sum of each row an each column. This will give us the marginal relative frequencies.

Now, the probability that a randomly selected male student prefers gym class is the probability that a student prefers gym class given that they are male. P(Gym|Male) This probability can be calculated using a conditional probability. P(Gym|Male) = P(Gymand Male)/P(Male) Looking at the table, we see that the probability that a random student is male and prefers gym equals 0.321. The total probability that a surveyed student is male is 0.472. Let's substitute the values. P(Gym|Male) = 0.321/0.472 We can now evaluate the ratio. P(Gym|Male) &= 0.321/0.472 & ≈ 0.680 The probability that a randomly selected male student prefers gym class is approximately 0.680, or 68 %.

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