Answered

As people exited a movie theater, they were informally surveyed about whether they enjoyed the movie or not.



Males:

Enjoyed: 47

Did not enjoy: 13



Females:

Enjoyed: 53

Did not enjoy: 3



\begin{table}

\captionsetup{labelformat=empty}

\caption{Survey on Movie}


EnjoyedDid Not EnjoyTotal

Males		a11\%

Females		46\%b

Total		100\%




\end{table}



What are the values of $a$ and $b$ in the relative frequency table for the survey results? Round answers to the nearest percent.


A$a = 41\%, b = 3\%$
B$a = 47\%, b = 3\%$
C$a = 40\%, b = 4\%$

Verified Answer :

  • Calculate the total number of participants: \(47 + 13 + 53 + 3 = 116\).
  • Determine the relative frequency \(a\) for 'Males who Enjoyed': \(\frac{47}{116} \times 100\% \approx 41\%\).
  • Determine the relative frequency \(b\) for 'Females who Did Not Enjoy': \(\frac{3}{116} \times 100\% \approx 3\%\).
  • Match the findings with the provided options, concluding the answer is \(\boxed{a = 41\%, b = 3\%}\).

Explanation

  1. Total Survey Participants Analysis

To find the missing relative frequencies \(a\) and \(b\) in the table, we first need to determine the total number of survey participants. Based on the data provided:

  • Males: \(47\) enjoyed the movie and \(13\) did not. Total males = \(47 + 13 = 60\).
  • Females: \(53\) enjoyed the movie and \(3\) did not. Total females = \(53 + 3 = 56\).
  • Total participants \(N = 60 + 56 = 116\).
  1. Calculating Relative Frequencies

The relative frequency of a group is defined as the number of people in that group divided by the total number of participants, expressed as a percentage.

  • For \(a\) (Males who Enjoyed):

\[a = \frac{47}{116} \times 100\% \approx 40.517\%\]
Rounding to the nearest percent, we get \(a = 41\%\).

  • For \(b\) (Females who Did Not Enjoy):

\[b = \frac{3}{116} \times 100\% \approx 2.586\%\]
Rounding to the nearest percent, we get \(b = 3\%\).

  1. Comparison and Final Conclusion

Comparing our calculated values with the given options:

  • Option a) \(a = 41\%, b = 3\%\)
  • Option b) \(a = 47\%, b = 3\%\)
  • Option c) \(a = 40\%, b = 4\%\)

Our calculated results, \(a = 41\%\) and \(b = 3\%\), match option (a). Thus, the missing values in the table are \(a = 41\%\) and \(b = 3\%\). The final answer is \(\boxed{a = 41\%, b = 3\%}\).

Examples

Understanding how to calculate relative frequencies is essential for interpreting data in many real-life situations. For example, if a store owner wants to know the demographic breakdown of their customers based on purchase behavior, they can calculate the percentage of each customer category relative to the total number of daily transactions. This helps businesses identify trends, such as which age group is most likely to buy a specific product, and make informed decisions to improve their services or marketing strategies. Effectively using such data enables better resource management and customer satisfaction.

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