To solve the given problem, we need to determine two things: the sample mean of the ratings and the proportion of ratings that are higher than the sample mean.
1. Calculate the Sample Mean:
- First, list all the ratings from the table:
[tex]\[
2, 5, 4, 3, 3, 3, 1, 3, 2, 1, 2, 4, 3, 3, 4, 4, 4, 5, 5, 4
\][/tex]
- Next, sum these ratings:
[tex]\[
2 + 5 + 4 + 3 + 3 + 3 + 1 + 3 + 2 + 1 + 2 + 4 + 3 + 3 + 4 + 4 + 4 + 5 + 5 + 4 = 64
\][/tex]
- There are 20 ratings in total.
- The sample mean is calculated by dividing the sum by the number of ratings, which is:
[tex]\[
\text{Sample Mean} = \frac{64}{20} = 3.25
\][/tex]
2. Determine the Proportion of Ratings Higher than the Sample Mean:
- Now we need to count how many ratings are higher than the sample mean (3.25).
- The ratings higher than 3.25 from the list are: 5, 4, 4, 4, 4, 5, 5, 4. There are 9 such ratings.
- To find the proportion, divide the number of ratings higher than the sample mean by the total number of ratings:
[tex]\[
\text{Proportion Higher} = \frac{9}{20} = 0.45
\][/tex]
Based on the calculations, the answers are as follows:
The sample mean is [tex]\( \boxed{3.25} \)[/tex].
The proportion of moviegoers in the sample who gave a rating higher than the sample mean is [tex]\( \boxed{0.45} \)[/tex].
