Questions: Consider a set of data in which the sample mean is 43.7 and the sample standard deviation is 3.1. Calculate the z-score given that x=49.6. Round your answer to two decimal places.

Consider a set of data in which the sample mean is 43.7 and the sample standard deviation is 3.1. Calculate the z-score given that x=49.6. Round your answer to two decimal places.

Transcript text: Consider a set of data in which the sample mean is 43.7 and the sample standard deviation is 3.1. Calculate the $z$-score given that $x=49.6$. Round your answer to two decimal places.

Solution

Solution Steps

To calculate the $z$-score, we use the formula: $ z = \frac{x - \mu}{\sigma} $, where $ x $ is the data point, $ \mu $ is the sample mean, and $ \sigma $ is the sample standard deviation. Substitute the given values into the formula and compute the result.

Step 1: Identify the Given Values

We are given the following values:

Sample mean ($\mu$): 43.7
Sample standard deviation ($\sigma$): 3.1
Data point ($x$): 49.6

Step 2: Apply the Z-Score Formula

The formula for calculating the $z$-score is: \[ z = \frac{x - \mu}{\sigma} \] Substitute the given values into the formula: \[ z = \frac{49.6 - 43.7}{3.1} \]

Step 3: Calculate the Z-Score

Perform the calculation: \[ z = \frac{5.9}{3.1} \approx 1.903 \]

Step 4: Round the Z-Score

Round the $z$-score to two decimal places: \[ z \approx 1.90 \]