Answered


Viewer's Age Group		ExcellentGoodAveragePoorMarginal Total

16-25		5242127113

26-35		33505997

36-45		58122834132

46-55		2517221276

56+		1253828

Marginal Total		1801267070446

\multicolumn{6}{|c|}{Note: A rating of good or excellent means the audience liked the movie, while a rating of poor means the audience disliked the movie.}






A movie producer conducted a survey after a preview screening of her new movie to find out how the film would be received by viewers from different age groups. The table shows the numbers of viewers in different age groups who rated the film excellent, good, average, and poor.



Out of the total respondents, the percentage of respondents from the 46-55 age group who rated the film excellent is [ ]%. Write your answer rounded to two decimal places.

Verified Answer :

  • Identify the number of respondents aged 46-55 who rated the film excellent: \(25\).
  • Identify the total number of respondents in the 46-55 age group: \(76\).
  • Divide the number of excellent ratings by the total respondents for the age group: \(\frac{25}{76} \approx 0.3289\).
  • Convert to a percentage by multiplying by 100 and rounding to two decimal places: \(\boxed{32.89\%}\).

Explanation

  1. Identify the relevant data from the contingency table

To determine the percentage of respondents in the 46-55 age group who rated the movie as 'Excellent', we first extract the relevant data from the provided contingency table. We focus on the row corresponding to the '46-55' age group. Looking at this row, we can see that:

  • The number of respondents in the 46-55 age group who rated the film as 'Excellent' is 25.
  • The total number of respondents in the 46-55 age group (the marginal total for that row) is 76.
  1. Set up the calculation for the percentage

To find the percentage of respondents in this specific age group who rated the film 'Excellent', we divide the number of 'Excellent' ratings in that age group by the total number of respondents in that same age group and then multiply by 100.
The formula is:
\[\text{Percentage} = \left( \frac{\text{Excellent in 46-55}}{\text{Total in 46-55}} \right) \times 100\]
Substituting the values:
\[\text{Percentage} = \left( \frac{25}{76} \right) \times 100\]

  1. Calculate and round the percentage

Performing the division:
\[\frac{25}{76} \approx 0.328947...\]
Now, we multiply by 100 to convert this decimal into a percentage:
\[0.328947... \times 100 = 32.8947...\%\]
Rounding the result to two decimal places, we get 32.89%.

Examples

This type of calculation is very useful for businesses and producers. For example, if you manage a chain of movie theaters, you could use this same method to see which specific demographic enjoys your films the most, helping you decide which movies to advertise to which age groups for better ticket sales! If \(\frac{25}{76}\) of viewers aged 46-55 loved a film, you know that this group is a strong target for similar future releases. The calculation is \(\frac{25}{76} \approx 0.3289 = 32.89\%\).

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