Suppose that the average waiting time for a patient at a physician's office is just over 29 minutes. To address the issue of long patient wait times, some physicians' offices are using wait-tracking systems to notify patients of expected wait times. Patients can adjust their arrival times based on this information and spend less time in waiting rooms. The following data show wait times (in minutes) for a sample of patients at offices that do not have a wait-tracking system and wait times for a sample of patients at offices with such systems.

Without Wait-
Tracking System	With Wait-Tracking
System
22	12
65	8
10	31
34	16
23	10
47	32
15	9
12	17
25	9
32	6
(a)	Considering only offices without a wait-tracking system, what is the z-score for the 10th patient in the sample (wait time = 32 minutes)?
If required, round your intermediate calculations and final answer to two decimal places.
z-score = 
 
(b)	Considering only offices with a wait-tracking system, what is the z-score for the 6th patient in the sample (wait time = 32 minutes)?
If required, round your intermediate calculations and final answer to two decimal places.
z-score =

To find the z-scores for the given wait times in both cases, we follow these steps:

The **z-score** formula is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \(x\) is the value for which we are calculating the z-score (the observed wait time),
- \(\mu\) is the mean of the data set,
- \(\sigma\) is the standard deviation of the data set.

### (a) For offices **without a wait-tracking system**:

We are given the wait time for the 10th patient: 32 minutes.

#### Step 1: Find the mean (\(\mu\)) of the wait times without a wait-tracking system.

The wait times are:
\[ 22, 65, 10, 34, 23, 47, 15, 12, 25, 32 \]

To calculate the mean:
\[
\mu = \frac{22 + 65 + 10 + 34 + 23 + 47 + 15 + 12 + 25 + 32}{10}
\]

#### Step 2: Calculate the standard deviation (\(\sigma\)).

The formula for the standard deviation is:
\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}
\]
Where:
- \(x_i\) are the individual wait times,
- \(n\) is the number of observations.

#### Step 3: Use the z-score formula for \(x = 32\).

### (b) For offices **with a wait-tracking system**:

We are given the wait time for the 6th patient: 32 minutes.

#### Step 1: Find the mean (\(\mu\)) of the wait times with a wait-tracking system.

The wait times are:
\[ 12, 8, 31, 16, 10, 32, 9, 17, 9, 6 \]

#### Step 2: Calculate the standard deviation (\(\sigma\)).

#### Step 3: Use the z-score formula for \(x = 32\).

Let me calculate both z-scores.Here are the results:

### (a) For offices **without a wait-tracking system**:
- Mean (\(\mu\)): 28.5 minutes
- Standard deviation (\(\sigma\)): 16.99 minutes
- Z-score for the 10th patient (wait time = 32 minutes):
  \[
  z = 0.21
  \]

### (b) For offices **with a wait-tracking system**:
- Mean (\(\mu\)): 15.0 minutes
- Standard deviation (\(\sigma\)): 9.35 minutes
- Z-score for the 6th patient (wait time = 32 minutes):
  \[
  z = 1.82
  \]

Would you like more details or have any other questions?

Here are 5 related questions:
1. What does a z-score tell us in terms of data distribution?
2. How can we interpret the z-score of 1.82 in the context of this problem?
3. How would adding more data points impact the mean and standard deviation?
4. What is the relationship between the mean and the median in a normally distributed dataset?
5. How can z-scores help identify outliers in a dataset?

**Tip:** A z-score closer to 0 indicates that the data point is near the mean, while higher absolute values indicate the point is further away from the mean.

Suppose that the average waiting time for a patient at a physician's office is just over 29 minutes. To address the issue of long patient wait times, some physicians' offices are using wait-tracking systems to notify patients of expected wait times. Patients can adjust their arrival times based on this information and spend less time in waiting rooms. The following data show wait times (in minutes) for a sample of patients at offices that do not have a wait-tracking system and wait times for a sample of patients at offices with such systems. Without Wait-Tracking System: 22, 65, 10, 34, 23, 47, 15, 12, 25, 32. With Wait-Tracking System: 12, 8, 31, 16, 10, 32, 9, 17, 9, 6. (a) Considering only offices without a wait-tracking system, what is the z-score for the 10th patient in the sample (wait time = 32 minutes)? (b) Considering only offices with a wait-tracking system, what is the z-score for the 6th patient in the sample (wait time = 32 minutes)?

This guide explains how to calculate the z-scores for patient wait times in physician offices with and without wait-tracking systems. Learn how to compute the mean and standard deviation to find the z-scores for specific patients. This example uses sample data to walk through the steps and formulas for accurate z-score calculation. Suitable for students in Grades 10-12.

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