https://file.aiquickdraw.com/mathbot/file/1726431930521vwcmcrkk.jpg

Let's address the two questions in the image:

### 4. Consider the dataset: 3, 4, 1, 2, 5. The Z-score of the number 5 is given by:

To calculate the Z-score, we use the formula:
\[
Z = \frac{x - \mu}{\sigma}
\]
where:
- \( x \) is the data point (in this case, 5),
- \( \mu \) is the mean of the dataset,
- \( \sigma \) is the standard deviation of the dataset.

#### Step 1: Calculate the mean (\( \mu \))
\[
\mu = \frac{3 + 4 + 1 + 2 + 5}{5} = \frac{15}{5} = 3
\]

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

First, find the squared differences from the mean:
- \( (3 - 3)^2 = 0 \)
- \( (4 - 3)^2 = 1 \)
- \( (1 - 3)^2 = 4 \)
- \( (2 - 3)^2 = 1 \)
- \( (5 - 3)^2 = 4 \)

Now, find the variance (average of squared differences):
\[
\text{Variance} = \frac{0 + 1 + 4 + 1 + 4}{5} = \frac{10}{5} = 2
\]
Then, the standard deviation is the square root of the variance:
\[
\sigma = \sqrt{2} \approx 1.41
\]

#### Step 3: Calculate the Z-score
Now, apply the Z-score formula for \( x = 5 \):
\[
Z = \frac{5 - 3}{1.41} = \frac{2}{1.41} \approx 1.42
\]

This is closest to **Option d) 1.58**.

---

### 5. Consider the following dataset: 17, 19, 36, 18, 9, 11, 21, 14, 16, 14. The number of outliers for this dataset is:

To identify the outliers, we use the interquartile range (IQR) method. Outliers are data points that fall below \( Q1 - 1.5 \times \text{IQR} \) or above \( Q3 + 1.5 \times \text{IQR} \).

#### Step 1: Arrange the data in ascending order
\[
9, 11, 14, 14, 16, 17, 18, 19, 21, 36
\]

#### Step 2: Find the quartiles
- \( Q1 \) (25th percentile) is the median of the lower half: \( Q1 = 14 \),
- \( Q3 \) (75th percentile) is the median of the upper half: \( Q3 = 19 \).

#### Step 3: Calculate the IQR
\[
\text{IQR} = Q3 - Q1 = 19 - 14 = 5
\]

#### Step 4: Find the outlier bounds
\[
\text{Lower bound} = Q1 - 1.5 \times \text{IQR} = 14 - 1.5 \times 5 = 14 - 7.5 = 6.5
\]
\[
\text{Upper bound} = Q3 + 1.5 \times \text{IQR} = 19 + 1.5 \times 5 = 19 + 7.5 = 26.5
\]

#### Step 5: Identify outliers
Any value below 6.5 or above 26.5 is considered an outlier. In the dataset, 36 is the only outlier.

Therefore, the number of outliers is **1** (Option b).

---

Would you like more details on these calculations or any clarification?

### Related Questions:
1. How do you calculate the Z-score for a negative data point?
2. What happens to the standard deviation if all data points in a dataset increase by the same constant?
3. How is variance related to standard deviation?
4. Can a dataset have multiple Z-scores for the same value?
5. How do different methods (e.g., IQR, Z-score) compare when identifying outliers?

### Tip:
When working with datasets, always organize the data in ascending order before calculating quartiles and IQR for outlier detection.

4) Consider the dataset: 3, 4, 1, 2, 5. The Z-score of the number 5 is given by:
5) Consider the following dataset: 17, 19, 36, 18, 9, 11, 21, 14, 16, 14. The number of outliers for this dataset is equal to:

Learn how to calculate the Z-score for the dataset [3, 4, 1, 2, 5] and determine the number of outliers in the dataset [17, 19, 36, 18, 9, 11, 21, 14, 16, 14]. This problem covers key statistical concepts such as Z-scores, standard deviation, and outlier detection using the IQR method.

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