Step-by-step Solution:

(a) Group the data into a frequency distribution with the first class as 60–69:

Data Set:
81, 82, 90, 94, 76, 72, 87, 68, 74, 66, 87, 92, 64, 88, 75, 62, 83, 77, 81, 96

1. Define the class intervals and count the frequencies:
   • Class interval: 60–69, 70–79, 80–89, 90–99
   • Tally the data points in each range.

Class Interval	Frequency (f)
60–69	4
70–79	5
80–89	8
90–99	3

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(b) Perform the calculations:

Step 1: Organize the data into a table with additional columns for calculations.

$x$	$x - \mu$	$(x - \mu)^2$

Where $\mu$ = mean.

**Step 1 Calculation numpy as np

Data

ages = np.array([81, 82, 90, 94, 76, 72, 87, 68, 74, 66, 87, 92, 64, 88, 75, 62, 83, 77, 81, 96])

Frequency Distribution

class_intervals = [(60, 69), (70, 79), (80, 89), (90, 99)] frequencies = [np.sum((ages >= interval[0]) & (ages <= interval[1])) for interval in class_intervals]

Mean

mean_age = np.mean(ages)

Variance

variance_age = np.var(ages, ddof=1) # ddof=1 for sample variance

Standard Deviation

std_dev_age = np.sqrt(variance_age)

Z-scores

z_scores = (ages - mean_age) / std_dev_age

Prepare results

frequencies, mean_age, variance_age, std_dev_age, z_scoresHere are the results of the calculations:

(a) Frequency distribution:

Class Interval	Frequency
60–69	4
70–79	5
80–89	7
90–99	4

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(b) Calculations:

(i) Mean:
$\mu = 79.75$

(ii) Variance:
$\text{Variance} = 102.20$

(iii) Standard Deviation:
$\text{SD} = 10.11$

(iv) Z-scores for each data point:

$Z = \frac{x - \mu}{\sigma}$

Z-scores:
$[ 0.12, 0.22, 1.01, 1.41, -0.37, -0.77, 0.72, -1.16, -0.57, -1.36, 0.72, 1.21, -1.56, 0.82, -0.47, -1.76, 0.32, -0.27, 0.12, 1.61 ]$

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(v) Correlation Coefficient:

This part requires two datasets to compute correlation, but as only one dataset is provided, clarification might be needed. Let me know if there's additional data for this computation.

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Would you like detailed steps for any specific calculation?

Related Questions:

1. How are Z-scores used in data analysis?
2. What does the standard deviation tell us about this dataset?
3. How can the variance be interpreted in a real-world context?
4. What are the advantages of grouping data into frequency intervals?
5. How does the mean help in understanding the dataset?

Tip: Always ensure your data is sorted and grouped appropriately before performing statistical analyses to avoid errors.