Question

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
survey on movie

	enjoyed	did not enjoy	total
females	46%	b

what are the values of a and b in the relative frequency table for the survey results? round answers to the nearest percent.
○ a = 40%, b = 4%
○ a = 41%, b = 3%
○ a = 47%, b = 3%
○ a = 41%, b = 19%

Explanation:

Step1: Calculate total number of males

Males who enjoyed: 47, did not enjoy: 13. Total males = \(47 + 13 = 60\).

Step2: Find relative frequency \(a\) for males who enjoyed

Relative frequency \(a=\frac{47}{60}\times100\%\approx78.33\%\)? Wait, no, wait. Wait, the table has females' enjoyed as 46%. Wait, total people: males (47+13=60), females (53+3=56). Wait, no, maybe total surveyed is males + females = 60 + 56 = 116? Wait, no, let's re - evaluate.

Wait, for females: enjoyed is 53, did not enjoy is 3. So total females = 53 + 3 = 56. The relative frequency of females who enjoyed is \(\frac{53}{total}\times100\% = 46\%\). So total surveyed \(total=\frac{53}{0.46}\approx115.22\approx115\) (wait, but 53/0.46 = 115.217...). Wait, maybe my approach is wrong.

Wait, another way: For males, did not enjoy is 13, and its relative frequency is 11%. Let total number of people be \(T\). Then \(\frac{13}{T}\times100\% = 11\%\), so \(T=\frac{13}{0.11}\approx118.18\approx118\).

Now, males who enjoyed: 47. So \(a=\frac{47}{118}\times100\%\approx39.83\%\approx40\%\)? No, wait, maybe the total for males is calculated from the 11% for did not enjoy. Wait, males: did not enjoy is 13, which is 11% of male total? Wait, maybe the table's "total" is per gender? Oh! That's the mistake. The table's total is per gender (males total and females total).

So for males: let male total be \(M\). Did not enjoy: 13, which is 11% of \(M\). So \(13 = 0.11M\), so \(M=\frac{13}{0.11}\approx118.18\approx118\). Then males who enjoyed: 47, so \(a=\frac{47}{118}\times100\%\approx39.83\%\approx40\%\)? No, wait, 47 + 13 = 60, so male total is 60. Then 13/60≈21.67%, but the table says 11%. So my initial assumption is wrong.

Wait, the table is a relative frequency table where the totals are the grand total (all people). Let's recast:

Let total number of people be \(N\).

Females: enjoyed = 53, did not enjoy = 3, so female total = 56. The relative frequency of females who enjoyed is 46%, so \(\frac{53}{N}=0.46\), so \(N = \frac{53}{0.46}\approx115.22\approx115\).

Males: enjoyed = 47, did not enjoy = 13, male total = 60.

Relative frequency of males who did not enjoy: \(\frac{13}{115}\times100\%\approx11.3\%\approx11\%\) (matches the table).

Now, relative frequency of males who enjoyed (\(a\)): \(\frac{47}{115}\times100\%\approx40.87\%\approx41\%\).

Relative frequency of females who did not enjoy (\(b\)): \(\frac{3}{115}\times100\%\approx2.61\%\approx3\%\).

So \(a\approx41\%\), \(b\approx3\%\).

Step1: Find total number of people using female enjoyed

Females who enjoyed: 53, relative frequency: 46% = 0.46. Let total \(N=\frac{53}{0.46}\approx115\).

Step2: Calculate \(a\) (males enjoyed relative frequency)

Males who enjoyed: 47. \(a=\frac{47}{115}\times100\%\approx41\%\).

Step3: Calculate \(b\) (females did not enjoy relative frequency)

Females who did not enjoy: 3. \(b=\frac{3}{115}\times100\%\approx3\%\).

Answer:

\(a = 41\%, b = 3\%\) (corresponding to the option \(a = 41\%, b = 3\%\))