Question

1. the following are weights of 14 young goats in pounds:

56 40 60 62 80 65 44 51 58 51
66 49 75 45
a) find the sample mean. round to one decimal place
b) find the median and the quartiles \\(q_1\\) and \\(q_3\\).

1. below are weights (in pounds) of male athletes in a high school.

weight frequency
130 - 150 13
150 - 170 18
170 - 190 23
190 - 210 32
210 - 230 11
estimate the mean for this data.

Explanation:

Calculate the sample mean for Question 1

We sum the 14 weights and divide by the sample size \(n = 14\).
The weights are: 56, 40, 60, 62, 80, 65, 44, 51, 58, 51, 66, 49, 75, 45.

$$ \sum x = 56 + 40 + 60 + 62 + 80 + 65 + 44 + 51 + 58 + 51 + 66 + 49 + 75 + 45 = 782 $$

$$ \bar{x} = \frac{782}{14} \approx 55.857 $$

Rounding to one decimal place gives \(55.9\).

Find the median and quartiles for Question 1

First, arrange the 14 data points in ascending order:
40, 44, 45, 49, 51, 51, 56, 58, 60, 62, 65, 66, 75, 80.

The median is the average of the 7th and 8th values:

$$ \text{Median} = \frac{56 + 58}{2} = 57 $$

For the quartiles, we split the ordered list into two halves:
Lower half (first 7 values): 40, 44, 45, 49, 51, 51, 56.
The median of this lower half is the 4th value:

$$ Q_1 = 49 $$

Upper half (last 7 values): 58, 60, 62, 65, 66, 75, 80.
The median of this upper half is the 4th value of this subset:

$$ Q_3 = 65 $$

Estimate the mean for grouped data in Question 2

We find the midpoint \(m_i\) for each class interval, multiply by its frequency \(f_i\), sum these products, and divide by the total frequency \(N\).

Weight Class	Midpoint (\(m_i\))	Frequency (\(f_i\))	\(f_i \times m_i\)
150 - 170	160	18	2880
170 - 190	180	23	4140
190 - 210	200	32	6400
210 - 230	220	11	2420

Sum of frequencies:

$$ N = 13 + 18 + 23 + 32 + 11 = 97 $$

Sum of products:

$$ \sum (f_i \times m_i) = 1820 + 2880 + 4140 + 6400 + 2420 = 17660 $$

Estimated mean:

$$ \mu \approx \frac{17660}{97} \approx 182.06 $$

Answer:

Question 1

a) The sample mean is 55.9 pounds.

b) The median is 57, the first quartile \(Q_1\) is 49, and the third quartile \(Q_3\) is 65.

Question 2

The estimated mean for this grouped data is approximately 182.1 pounds (or 182.06 pounds).