Question

part 2: biggie size those fries.... or maybe not!!!

	small fries (calories)	medium fries (calories)	large fries (calories)
wendys	330	410	540
burger king	340	440	540
dairy queen		310	500
sonic	204	326	448
steak and shake	240	440	640
chick-fil-a	290	380	430
arby’s	410	540	640

1. write a statistical question that can be answered by the data in the table above.

1. find the 5 number summary for each size of fries:

Explanation:

Question 1:

A statistical question should involve variability and be answerable by analyzing the data. For example, we can ask about the average calories of a certain size of fries or the range of calories for a specific size across different restaurants. A suitable question could be: "What is the average number of calories in a large fries across all these restaurants?" or "What is the range of calories for small fries among these fast - food chains?" (Here we'll take "What is the average number of calories in a large fries across all these restaurants?" as an example)

Answer:

What is the average number of calories in a large fries across all these restaurants? (Other valid statistical questions like "What is the median number of calories for medium fries?" or "Which restaurant has the highest calorie count for large fries?" are also acceptable)

Question 2:

We'll find the 5 - number summary (minimum, first quartile \(Q_1\), median, third quartile \(Q_3\), maximum) for each size of fries.

Small Fries:

First, list the calorie values for small fries: 230 (McDonalds), 330 (Wendys), 340 (Burger King), 204 (Sonic), 240 (Steak and Shake), 290 (Chick - fil - A), 410 (Arby's). Wait, Dairy Queen has no value for small fries, so we exclude it. Now sort the data: 204, 230, 240, 290, 330, 340, 410.

Step 1: Find the minimum

The smallest value in the sorted data is 204.

Step 2: Find the median (second quartile, \(Q_2\))

There are 7 data points. The median is the \((\frac{7 + 1}{2})=4^{th}\) value. So the median is 290.

Step 3: Find \(Q_1\) (first quartile)

The first quartile is the median of the lower half of the data. The lower half is 204, 230, 240 (excluding the median). The median of these 3 values is the \(2^{nd}\) value, so \(Q_1 = 230\).

Step 4: Find \(Q_3\) (third quartile)

The third quartile is the median of the upper half of the data. The upper half is 330, 340, 410 (excluding the median). The median of these 3 values is the \(2^{nd}\) value, so \(Q_3=340\).

Step 5: Find the maximum

The largest value in the sorted data is 410.
So the 5 - number summary for small fries is: Minimum = 204, \(Q_1 = 230\), Median = 290, \(Q_3 = 340\), Maximum = 410.

Medium Fries:

List the calorie values for medium fries: 380 (McDonalds), 410 (Wendys), 440 (Burger King), 310 (Dairy Queen), 326 (Sonic), 440 (Steak and Shake), 380 (Chick - fil - A), 540 (Arby's). Sort the data: 310, 326, 380, 380, 410, 440, 440, 540.

Step 1: Find the minimum

The smallest value is 310.

Step 2: Find the median (\(Q_2\))

There are 8 data points. The median is the average of the \(\frac{8}{2}=4^{th}\) and \((\frac{8}{2}+ 1)=5^{th}\) values. So median=\(\frac{380 + 410}{2}=\frac{790}{2}=395\).

Step 3: Find \(Q_1\)

The first quartile is the median of the lower half (310, 326, 380, 380). There are 4 data points, so the median is the average of the \(2^{nd}\) and \(3^{rd}\) values. \(Q_1=\frac{326+380}{2}=\frac{706}{2} = 353\).

Step 4: Find \(Q_3\)

The third quartile is the median of the upper half (410, 440, 440, 540). There are 4 data points, so the median is the average of the \(2^{nd}\) and \(3^{rd}\) values. \(Q_3=\frac{440 + 440}{2}=440\).

Step 5: Find the maximum

The largest value is 540.
So the 5 - number summary for medium fries is: Minimum = 310, \(Q_1 = 353\), Median = 395, \(Q_3 = 440\), Maximum = 540.

Large Fries:

List the calorie values for large fries: 500 (McDonalds), 540 (Wendys), 540 (Burger King), 500 (Dairy Queen), 448 (Sonic), 640 (Steak and Shake), 430 (Chick - fil - A), 640 (Arby's). Sort the data: 430, 448, 500, 500, 540, 540, 640, 640.

Step 1: Find the minimum

The smallest value is 430.

Step 2: Find the median (\(Q_2\))

There are 8 data points. The median is the average of the \(4^{th}\) and \(5^{th}\) values. So median=\(\frac{500+540}{2}=\frac{1040}{2}=520\).

Step 3: Find \(Q_1\)

The first quartile is the median of the lower half (430, 448, 500, 500). There are 4 data points, so the median is the average of the \(2^{nd}\) and \(3^{rd}\) values. \(Q_1=\frac{448 + 500}{2}=\frac{948}{2}=474\).

Step 4: Find \(Q_3\)

The third quartile is the median of the upper half (540, 540, 640, 640). There are 4 data points, so the median is the average of the \(2^{nd}\) and \(3^{rd}\) values. \(Q_3=\frac{540+640}{2}=\frac{1180}{2}=610\).

Step 5: Find the maximum

The largest value is 640.
So the 5 - number summary for large fries is: Minimum = 430, \(Q_1 = 474\), Median = 520, \(Q_3 = 610\), Maximum = 640.