Question

the following are the amounts of total fat (in grams) in different kinds of sweet treats available at the local donut shop.

24 16 22 18 17 21 15 15 24 21
17 23 18 17 17 23 23 19 15 15

a. what is the range for this data set?

grams

b. what is the standard deviation for this data set? round your answer to the nearest tenth, if necessary.

grams

Explanation:

Part a: Range Calculation

Step1: Find Max and Min Values

First, identify the maximum and minimum values in the data set. The data points are: 24, 16, 22, 18, 17, 21, 15, 15, 24, 21, 17, 23, 18, 17, 17, 23, 23, 19, 15, 15.
The maximum value ($\text{Max}$) is 24, and the minimum value ($\text{Min}$) is 15.

Step2: Calculate Range

The range is calculated as $\text{Range} = \text{Max} - \text{Min}$.
Substituting the values: $\text{Range} = 24 - 15 = 9$.

Step1: Calculate the Mean ($\mu$)

First, sum all the data points. Let's list the data and count the number of values ($n$).
Data: 24, 16, 22, 18, 17, 21, 15, 15, 24, 21, 17, 23, 18, 17, 17, 23, 23, 19, 15, 15.
Count ($n$) = 20.

Sum of data:
$24 + 16 + 22 + 18 + 17 + 21 + 15 + 15 + 24 + 21 + 17 + 23 + 18 + 17 + 17 + 23 + 23 + 19 + 15 + 15$
Let's compute step-by-step:
$24+16=40$; $40+22=62$; $62+18=80$; $80+17=97$; $97+21=118$; $118+15=133$; $133+15=148$; $148+24=172$; $172+21=193$; $193+17=210$; $210+23=233$; $233+18=251$; $251+17=268$; $268+17=285$; $285+23=308$; $308+23=331$; $331+19=350$; $350+15=365$; $365+15=380$.

Mean: $\mu = \frac{\text{Sum}}{n} = \frac{380}{20} = 19$.

Step2: Calculate Squared Differences from Mean

For each data point $x_i$, compute $(x_i - \mu)^2$:

• $24$: $(24 - 19)^2 = 25$
• $16$: $(16 - 19)^2 = 9$
• $22$: $(22 - 19)^2 = 9$
• $18$: $(18 - 19)^2 = 1$
• $17$: $(17 - 19)^2 = 4$ (appears 4 times: total $4 \times 4 = 16$)
• $21$: $(21 - 19)^2 = 4$ (appears 2 times: total $2 \times 4 = 8$)
• $15$: $(15 - 19)^2 = 16$ (appears 4 times: total $4 \times 16 = 64$)
• $24$: $(24 - 19)^2 = 25$ (appears 2 times: total $2 \times 25 = 50$)
• $23$: $(23 - 19)^2 = 16$ (appears 3 times: total $3 \times 16 = 48$)
• $19$: $(19 - 19)^2 = 0$

Now sum all these squared differences:
$25 + 9 + 9 + 1 + 16 + 8 + 64 + 50 + 48 + 0$

Compute step-by-step:
$25+9=34$; $34+9=43$; $43+1=44$; $44+16=60$; $60+8=68$; $68+64=132$; $132+50=182$; $182+48=230$; $230+0=230$.

Step3: Calculate Variance ($\sigma^2$)

Variance is the average of the squared differences: $\sigma^2 = \frac{\text{Sum of squared differences}}{n} = \frac{230}{20} = 11.5$.

Step4: Calculate Standard Deviation ($\sigma$)

Standard deviation is the square root of the variance: $\sigma = \sqrt{11.5} \approx 3.4$ (rounded to the nearest tenth).

Answer:

9

Part b: Standard Deviation Calculation