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Chapter 3: Problem 14

The tuition costs (in dollars) for a sample of four-year state colleges in California and Texas are shown below. Compare the means and the standard deviations of the data and compare the state tuition costs of the two states in a sentence or two. CA: \(7040,6423,6313,6802,7048,7460\) TX: \(7155,7504,7328,8230,7344,5760\)

Short Answer

Expert verified
The mean cost of tuition is higher in Texas (\$7220.2) compared to California (\$6847.7). The standard deviation, a measure of variability, is also higher in Texas (817.52) which means their costs vary more. This suggests that the cost of state colleges in Texas can be more unpredictable compared to California.

Step by step solution

01

Calculate the Mean

To calculate the mean (average), add up all the values and then divide by the number of values. For CA, the mean is \(\frac{7040 + 6423 + 6313 + 6802 + 7048 + 7460}{6} = 6847.7\). For TX, the mean is \(\frac{7155 + 7504 + 7328 + 8230 + 7344 + 5760}{6} = 7220.2\)
02

Calculate the Standard Deviation

To find the standard deviation, first calculate the variance by subtracting the mean from each value, squaring the results, summing them, and dividing by the number of values. The standard deviation is the square root of the variance. The variance for CA is \(\frac{(7040-6847.7)^2 + (6423-6847.7)^2 + (6313-6847.7)^2 + (6802-6847.7)^2 + (7048-6847.7)^2 + (7460-6847.7)^2}{6} = 267771.5\) and thus the standard deviation is \(\sqrt{267771.5} \approx 517.47\). Using same process, TX's standard deviation is calculated to be \(\sqrt{668150.2}\) approximately equals to 817.52.
03

Compare the Two States' Tuition Costs

Compare the mean and standard deviation of both states. The comparison should be articulated in clear sentences.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean calculation
The mean is a measure of the central tendency of a data set. It tells you what the average value is for a set of numbers. Calculating the mean is straightforward. Here's how you can do it:
  • Add up all the values in your data set.
  • Divide by the number of values you added.
For example, when looking at the tuition costs for colleges in California, you would add: 7040, 6423, 6313, 6802, 7048, and 7460. Then, divide the total by 6, which gives you a mean of 6847.7 dollars. In Texas, adding up the numbers 7155, 7504, 7328, 8230, 7344, and 5760 and dividing by 6 gives you a mean of 7220.2 dollars.

Keep in mind, the mean provides a basic overview of your data, but it doesn't explain how much your data points vary.
Standard deviation
Standard deviation helps determine the spread of a data set. It's essentially a number that tells you how much, on average, each number in a set differs from the mean. A smaller standard deviation indicates data points tend to be close to the mean, while a larger one indicates data points are spread out over a larger range of values.

To calculate standard deviation, first find the variance:
  • Subtract the mean from each data point and square the result.
  • Add all the squared results together.
  • Divide by the number of data points to find the variance.
Finally, take the square root of the variance to get the standard deviation.

Using this method, California's variance is 267771.5 and its standard deviation is approximately 517.47. For Texas, the standard deviation is about 817.52; thus, indicating a larger dispersion.
Variance
Variance is a key concept in statistics that measures the variability of a data set. It tells you how spread out the numbers in your data set are. Think of it as a stepping stone to finding the standard deviation. Here鈥檚 how it works:
  • Calculate the mean of your data set.
  • Subtract the mean from each data point to find the deviation of each value.
  • Square each deviation to eliminate negative values.
  • Average these squared deviations to find the variance.
Variance for California's tuition is calculated as 267771.5, while for Texas, it stands at 668150.2. These figures show that Texas has a greater variability in tuition costs compared to California.

Understanding variance is pivotal because it serves as the foundation for calculating standard deviation, which is more intuitive in explaining data spread.
Data comparison
When comparing data sets, like the tuition costs of colleges in California and Texas, you're looking at both the mean and standard deviation for insights. Here's what's important:
  • Look at the mean to see which state has higher or lower average tuition.
  • Consider the standard deviation to understand the variability within the tuition costs.
For California, the mean tuition is 6847.7 with a standard deviation of 517.47, suggesting relatively consistent tuition fees across colleges. Contrastingly, Texas shows a higher mean of 7220.2 and a larger standard deviation of 817.52, indicating greater variability.

With statistics, you're not just looking at numbers - you're unlocking the story they tell. Comparing means and standard deviations helps provide a comprehensive picture of what's happening in the data.

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Most popular questions from this chapter

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