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11.5

Expert-verifiedFound in: Page 634

Book edition
OER 2018

Author(s)
Barbara Illowsky, Susan Dean

Pages
902 pages

ISBN
9781938168208

A sample of $300$students is taken. Of the students surveyed,$50$ were music students, while $250$were not. Ninetyseven were on the honor roll, while $203$ were not. If we assume being a music student and being on the honor roll are independent events, what is the expected number of music students who are also on the honor roll?

Around $16$ kids from a sample are likely to be music students who are also on the honour roll.

Given in the question that, A sample of $300$ students is taken. Of the students surveyed, $50$ were music students, while $250$ were not. Ninetyseven were on the honor roll, while $203$were not. If we assume being a music student and being on the honor roll are independent events. We need to find the expected number of music students who are also on the honor roll if we assume being a music student and being on the honor roll are independent events.

A total of $300$ pupils are chosen as a sample. $50$ of the students polled were music students, whereas the other $250$were not. Ninety-seven students were on the honour roll, while the remaining $203$ were not.

Calculation:

Let the random variable $X$be defined as

$X=$ Expected number of students on the honour roll who are also music students.

Let$A$ represent the fact that the student is a music student, and $B$represent the fact that he or she is also on honour roll. We have the following condition if both occurrences $A$ and$B$ are independent events:

$P(AANDB)=P\left(A\right)\xb7P\left(B\right)$

We have,

$P\left(A\right)=\frac{50}{300}$

$P\left(B\right)=\frac{97}{300}$

$\Rightarrow \frac{X}{300}=\frac{50}{300}\times \frac{97}{300}$

localid="1653556696458" $\Rightarrow X=\frac{4850}{300}\phantom{\rule{0ex}{0ex}}=16.17\phantom{\rule{0ex}{0ex}}\approx 16$

$\therefore X=16$

A sample of $212$ commercial businesses was surveyed for recycling one commodity; a commodity here means any one type of recyclable material such as plastic or aluminum. Table $11.41$ shows the business categories in the survey, the sample size of each category, and the number of businesses in each category that recycle one commodity. Based on the study, on average half of the businesses were expected to be recycling one commodity. As a result, the last column shows the expected number of businesses in each category that recycle one commodity. At the $5$% significance level, perform a hypothesis test to determine if the observed number of businesses that recycle one commodity follows the uniform distribution of the expected values.

Business Type | Number in class | Observed Number that recycles one commodity | Expected number that recycles one commodity |

office | $35$ | $19$ | $17.5$ |

Retail/Wholesale | $48$ | $27$ | $24$ |

Food/Restaurants | $53$ | $35$ | $26.5$ |

Manufacturing/Medical | $52$ | $21$ | $26$ |

Hotel/Mixed | $24$ | $9$ | $12$ |

**Table 11.41**

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