The total income for white households making under 250,000...

The total income for white households making under $\$ 250,000$ in 2008 is given in the following table. Source: U.S. Census Bureau. a. Use this table to estimate the mean income for white households in 2008. b. Compare this estimate with the estimate found in Exercise 39. Discuss whether this provides evidence that white American households have higher earnings than African American households.

The total income for white households making under $\$ 250,000$ in 2008 is given in the following table. Source: U.S. Census Bureau.
a. Use this table to estimate the mean income for white households in 2008.
b. Compare this estimate with the estimate found in Exercise 39. Discuss whether this provides evidence that white American households have higher earnings than African American households.

Key Concepts

Between-Group Comparison

Comparing statistical estimates between different groups involves analyzing the central tendency measures, such as the mean, of each group. These comparisons help in inferring whether one group tends to have higher or lower values than another, but they need to be interpreted with caution considering potential variability and factors like sample representation.

Interpreting Statistical Evidence

Assessing whether differences in estimated means provide sufficient evidence for one group having higher earnings than another requires more than just a comparison of averages. It involves considering factors like the variability within each group, the sample sizes, and whether the observed differences are statistically significant, thereby supporting a well-grounded inference.

Grouped Data Analysis

When data is provided in groups (or intervals) rather than as individual observations, estimating measures like the mean requires representing each group by a typical value—usually the midpoint—and then using the frequency (or weight) of that group in calculations. This method provides an approximate average when complete raw data is not available.

Weighted Mean Calculation

The mean of a dataset can be estimated by multiplying each group's representative value (often the midpoint) by the frequency of observations in that group, summing these products, and then dividing by the total number of observations. This approach ensures that groups with more data influence the average more than those with less data.

Transcript

00:01 This question shows us a table of the frequency table of incomes for white families making under $250 ,000 a year.

00:11 Now, we know what we're finding, estimating a mean using a frequency table.

00:17 We have that the mean, x bar, is equal to the sum of the value times the frequency over the sum of the frequency.

00:27 Now, here we have ranges as our values.

00:31 So that means we need to take the midpoint.

00:34 Luckily, the midpoint is actually already included in this table, so we're good.

00:39 So i'm going to copy down part of this table, and we'll add on to it and do some calculations from there.

00:45 So for x, our value, i'm not going to write down the whole number.

00:49 I'm just going to write down the number in thousands of dollars.

00:52 So first we'll have 10, then 30, then 50, 70, 19, 125, 175, and 225.

01:07 Then we have our frequency.

01:11 And now for frequency, i'll write down the whole thing.

01:14 This is already in thousands of people, so it'll be 16 ,241, which is really 16 ,0241 ,000.

01:24 That will have 20 ,487 ,000, 16 ,065 ,000, 12 ,7 ,000, 12 ,7 ,7 ,000, 12 ,7 ,7 ,7 ,000, 12 ,7 ,7 ,747 ,000 ,12 ,12 ,7 ,7 ,747 ,000.

01:36 $6 ,000 ,9 ,080 ,000, 12 ,230 ,4 ,4 ,498, and finally, 1 ,743 ,000.

01:56 So next i'm going to make a column called x times f, and i'll just multiply these two columns together to give us the thing that we need to sum up.

02:04 X times f will get 162 ,410, then 614 ,610, 803 ,250, then 892, 220, 817 ,000, 200, 1 ,528 ,750, 787 ,150 ,000, 150...

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