Problem

A highly anticipated sci-fi movie opened in theaters with a special midnight showing this past Friday. For the Saturday edition of her town's newspaper, Salma wrote a column about local interest in the movie. She found that the attendance for the midnight showing at her town's theater was 221 people. For all the theaters in the state, attendance at the midnight showing had a mean of 245 people with a standard deviation of 37 people. Find the $z$-score of the attendance at the theater in Salma's town relative to the attendances among all the theaters in the state. Round your answer to two decimal places. \[ z=\square \]

Solution

Step 1 :A highly anticipated sci-fi movie opened in theaters with a special midnight showing this past Friday. For the Saturday edition of her town's newspaper, Salma wrote a column about local interest in the movie. She found that the attendance for the midnight showing at her town's theater was 221 people. For all the theaters in the state, attendance at the midnight showing had a mean of 245 people with a standard deviation of 37 people.

Step 2 :We are asked to find the $z$-score of the attendance at the theater in Salma's town relative to the attendances among all the theaters in the state.

Step 3 :The $z$-score is calculated by subtracting the mean from the observed value, and then dividing by the standard deviation. In this case, the observed value is the attendance at Salma's town's theater, which is 221 people. The mean is the average attendance at all the theaters in the state, which is 245 people. The standard deviation is 37 people.

Step 4 :Substituting these values into the formula, we get: \(z = \frac{X - \mu}{\sigma}\)

Step 5 :Substituting the given values, we get: \(z = \frac{221 - 245}{37}\)

Step 6 :Solving the above expression, we get: \(z = -0.6486486486486487\)

Step 7 :Rounding to two decimal places, we get the final answer: \(\boxed{-0.65}\)

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Source: https://solvelyapp.com/problems/ZZigOKHjoR/

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