Question

1. the following data represents a sample of senior students in a high - school english course who were asked the question: \how many books have you read since your freshman year of high school?\ their answers: 75, 83, 96, 100, 121, 125 a) find and interpret the range. b) find and interpret the standard deviation. show your work!

Explanation:

Step1: Recall range formula

The range of a data - set is given by $R = \text{Max}-\text{Min}$, where $\text{Max}$ is the maximum value and $\text{Min}$ is the minimum value in the data - set.

Step2: Identify Max and Min values

In the data - set $\{75,83,96,100,121,125\}$, $\text{Min}=75$ and $\text{Max}=125$.

Step3: Calculate the range

$R=125 - 75=50$.
Interpretation: The range represents the difference between the highest and the lowest number of books read by the senior students. It shows that the number of books read by the most avid reader in the sample is 50 more than the number of books read by the least avid reader in the sample.

Step4: Recall standard deviation formula

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $x_{i}$ are the data points, $\bar{x}$ is the sample mean, and $n$ is the sample size.

Step5: Calculate the sample mean

$\bar{x}=\frac{75 + 83+96+100+121+125}{6}=\frac{600}{6}=100$.

Step6: Calculate $(x_{i}-\bar{x})^{2}$ for each data point

For $x_1 = 75$: $(75 - 100)^{2}=(-25)^{2}=625$.
For $x_2 = 83$: $(83 - 100)^{2}=(-17)^{2}=289$.
For $x_3 = 96$: $(96 - 100)^{2}=(-4)^{2}=16$.
For $x_4 = 100$: $(100 - 100)^{2}=0$.
For $x_5 = 121$: $(121 - 100)^{2}=21^{2}=441$.
For $x_6 = 125$: $(125 - 100)^{2}=25^{2}=625$.

Step7: Calculate $\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}$

$\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=625+289+16+0+441+625 = 1996$.

Step8: Calculate the sample standard deviation

$s=\sqrt{\frac{1996}{6 - 1}}=\sqrt{\frac{1996}{5}}=\sqrt{399.2}\approx20$.
Interpretation: The standard deviation measures the average amount by which each data point differs from the mean. A standard deviation of approximately 20 means that, on average, the number of books read by each student in the sample differs from the mean number of books read (100) by about 20 books.

Answer:

a) The range is 50. Interpretation: It shows the difference between the highest and lowest number of books read in the sample.
b) The standard deviation is approximately 20. Interpretation: It shows the average deviation of the number of books read by each student from the mean number of books read.