Let's break this problem down step by step.

1. Sample Mean (μ):

The sample mean is the sum of the data points divided by the number of data points.

$\mu = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{1.6 + 5.2 + 2.8 + 3.7 + 4.0}{5}$

2. Sample Standard Deviation (s):

The formula for sample standard deviation is:

$s = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}}$

where $x_i$ are the data points, $\mu$ is the sample mean, and $n$ is the number of data points.

3. Z-Score for 2.8:

The z-score is calculated using:

$z = \frac{x - \mu}{s}$

where $x$ is the data point (2.8 in this case), $\mu$ is the sample mean, and $s$ is the sample standard deviation.

4. Percentile Rank for 2.8:

To calculate the percentile rank of a value, we determine how many data points are below it and then use the formula:

$\text{Percentile Rank} = \frac{\text{Number of values below 2.8}}{n} \times 100$

Let me first calculate each of these step-by-step for you. numpy as np

Data

data = [1.6, 5.2, 2.8, 3.7, 4.0]

1. Sample Mean

mean = np.mean(data)

2. Sample Standard Deviation

std_dev = np.std(data, ddof=1)

3. Z-score for 2.8

z_score_2_8 = (2.8 - mean) / std_dev

4. Percentile rank for 2.8

below_2_8 = sum(x < 2.8 for x in data) percentile_rank_2_8 = (below_2_8 / len(data)) * 100

(mean, round(std_dev, 2), round(z_score_2_8, 2), round(percentile_rank_2_8, 2))Here are the results based on the calculations:

1. Sample Mean: $3.46$
2. Sample Standard Deviation: $1.35$ (rounded to 2 decimals)
3. Z-Score for 2.8: $-0.49$ (rounded to 2 decimals)
4. Percentile Rank for 2.8: $20.0\%$

Would you like further explanations or help with anything else?

Here are 5 related questions you can explore:

1. How do you interpret a negative z-score?
2. What does a percentile rank of 20% mean in this context?
3. How can the sample mean change with additional data points?
4. What is the difference between a sample and population standard deviation?
5. How does the z-score relate to the normal distribution?

Tip: A negative z-score indicates that the value is below the sample mean, while a positive z-score means it's above the mean.