Question

1. airline fares can vary greatly, even from the same carrier within the same day. the following are fares from new york to burbank, ca, over the past week. $430, $567, $334, $701, $424, $555, $890, $455, $450, $1,122 what is the standard deviation for this distribution? round your answers to the nearest dollar. 5. the distribution of cell phone bills for families in smithtown north high school has mean $183 and standard deviation 11. at smithtown south high school, the mean is $181 and the standard deviation is 21. which distribution is more spread out?

Explanation:

Question 4

Step 1: Calculate the mean (\(\mu\))

First, we find the sum of the data points. The data points are \(430, 567, 334, 701, 424, 555, 890, 455, 450, 1122\).
The sum \( \sum x = 430 + 567 + 334 + 701 + 424 + 555 + 890 + 455 + 450 + 1122 \)
\( \sum x = 430+567 = 997; 997 + 334 = 1331; 1331+701 = 2032; 2032+424 = 2456; 2456+555 = 3011; 3011+890 = 3901; 3901+455 = 4356; 4356+450 = 4806; 4806+1122 = 5928 \)
There are \(n = 10\) data points. So the mean \( \mu=\frac{\sum x}{n}=\frac{5928}{10} = 592.8\)

Step 2: Calculate the squared differences from the mean

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

• For \(x = 430\): \((430 - 592.8)^2=(-162.8)^2 = 26503.84\)
• For \(x = 567\): \((567 - 592.8)^2=(-25.8)^2 = 665.64\)
• For \(x = 334\): \((334 - 592.8)^2=(-258.8)^2 = 66977.44\)
• For \(x = 701\): \((701 - 592.8)^2=(108.2)^2 = 11707.24\)
• For \(x = 424\): \((424 - 592.8)^2=(-168.8)^2 = 28493.44\)
• For \(x = 555\): \((555 - 592.8)^2=(-37.8)^2 = 1428.84\)
• For \(x = 890\): \((890 - 592.8)^2=(297.2)^2 = 88327.84\)
• For \(x = 455\): \((455 - 592.8)^2=(-137.8)^2 = 18988.84\)
• For \(x = 450\): \((450 - 592.8)^2=(-142.8)^2 = 20401.44\)
• For \(x = 1122\): \((1122 - 592.8)^2=(529.2)^2 = 279952.64\)

Step 3: Calculate the sum of squared differences

\(\sum (x_i - \mu)^2=26503.84 + 665.64+66977.44 + 11707.24+28493.44 + 1428.84+88327.84 + 18988.84+20401.44 + 279952.64\)
Let's add them step by step:
\(26503.84+665.64 = 27169.48\); \(27169.48+66977.44 = 94146.92\); \(94146.92+11707.24 = 105854.16\); \(105854.16+28493.44 = 134347.6\); \(134347.6+1428.84 = 135776.44\); \(135776.44+88327.84 = 224104.28\); \(224104.28+18988.84 = 243093.12\); \(243093.12+20401.44 = 263494.56\); \(263494.56+279952.64 = 543447.2\)

Step 4: Calculate the variance (\(\sigma^2\))

Variance for a population (since we have all the data of the past week, we assume it's a population) is \(\sigma^2=\frac{\sum (x_i - \mu)^2}{n}\)
\(\sigma^2=\frac{543447.2}{10}=54344.72\)

Step 5: Calculate the standard deviation (\(\sigma\))

Standard deviation is the square root of variance: \(\sigma=\sqrt{54344.72}\approx 233.12\approx 233\) (rounded to the nearest dollar)

The standard deviation measures the spread of a distribution. A larger standard deviation means the data is more spread out. For Smithtown North High School, the standard deviation is 11, and for Smithtown South High School, it is 21. Since \(21> 11\), the distribution of cell phone bills for families in Smithtown South High School is more spread out.

Answer:

The standard deviation is \(\$233\)

Question 5