Math / Data & Statistics

QuestionRecently, a random sample of 25-34 year olds was asked, "How much do you currently have in savings, not including retirement savings?" The data in the table represent the responses to the survey. Approximate the mean and standard deviation amount of savings.
Click the icon to view the frequency distribution for the amount of savings.
The sample mean amount of savings is $\$$ $\square$ (Round to the nearest dollar as needed.) Frequency distribution of amount of savings \begin{tabular}{cc} Savings & Frequency \\ \hline $\$ 0-\$ 199$ & 348 \\ \hline $\$ 200-\$ 399$ & 96 \\ \hline $\$ 400-\$ 599$ & 48 \\ \hline $\$ 600-\$ 799$ & 23 \\ \hline $\$ 800-\$ 999$ & 11 \\ \hline $\$ 1000-\$ 1199$ & 6 \\ \hline $\$ 1200-\$ 1399$ & 1 \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The data is presented in a grouped frequency distribution.
2. We assume the savings amounts are uniformly distributed within each range.
3. The midpoint of each savings range represents the typical value for that range.
4. The mean and standard deviation can be approximated using the midpoints and frequencies of the ranges.

STEP 2

1. Calculate the midpoint for each range of savings.
2. Use the midpoints and frequencies to approximate the mean savings.
3. Use the midpoints, frequencies, and the mean to approximate the standard deviation.

STEP 3

Calculate the midpoint for the first range $\$0 - \$199$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{0 + 199}{2} = 99.5$

STEP 4

Calculate the midpoint for the second range $\$200 - \$399$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{200 + 399}{2} = 299.5$

STEP 5

Calculate the midpoint for the third range $\$400 - \$599$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{400 + 599}{2} = 499.5$

STEP 6

Calculate the midpoint for the fourth range $\$600 - \$799$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{600 + 799}{2} = 699.5$

STEP 7

Calculate the midpoint for the fifth range $\$800 - \$999$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{800 + 999}{2} = 899.5$

STEP 8

Calculate the midpoint for the sixth range $\$1000 - \$1199$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{1000 + 1199}{2} = 1099.5$

STEP 9

Calculate the midpoint for the seventh range $\$1200 - \$1399$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{1200 + 1399}{2} = 1299.5$

STEP 10

Calculate the total number of respondents by summing the frequencies.
$\text{Total Frequency} = 348 + 96 + 48 + 23 + 11 + 6 + 1 = 533$

STEP 11

Calculate the weighted sum of the midpoints using the frequencies.
$\text{Weighted Sum} = 99.5 \times 348 + 299.5 \times 96 + 499.5 \times 48 + 699.5 \times 23 + 899.5 \times 11 + 1099.5 \times 6 + 1299.5 \times 1$

STEP 12

Perform the calculations for each term in the weighted sum.
$99.5 \times 348 = 34626$ $299.5 \times 96 = 28752$ $499.5 \times 48 = 23976$ $699.5 \times 23 = 16088.5$ $899.5 \times 11 = 9894.5$ $1099.5 \times 6 = 6597$ $1299.5 \times 1 = 1299.5$

STEP 13

Sum the results from the previous step to find the total weighted sum.
$\text{Total Weighted Sum} = 34626 + 28752 + 23976 + 16088.5 + 9894.5 + 6597 + 1299.5 = 120233.5$

STEP 14

Approximate the mean savings by dividing the total weighted sum by the total frequency.
$\text{Mean} = \frac{120233.5}{533} \approx 225.58$ Rounding to the nearest dollar: $\text{Mean} \approx 226$

STEP 15

Calculate the squared differences between each midpoint and the mean, weighted by the frequency.
$(99.5 - 225.58)^2 \times 348$ $(299.5 - 225.58)^2 \times 96$ $(499.5 - 225.58)^2 \times 48$ $(699.5 - 225.58)^2 \times 23$ $(899.5 - 225.58)^2 \times 11$ $(1099.5 - 225.58)^2 \times 6$ $(1299.5 - 225.58)^2 \times 1$

STEP 16

Perform the calculations for each term in the weighted squared differences.
$(99.5 - 225.58)^2 \times 348 = 15805.2764 \times 348 = 5498988.472$ $(299.5 - 225.58)^2 \times 96 = 5457.2464 \times 96 = 523899.654$ $(499.5 - 225.58)^2 \times 48 = 75189.5764 \times 48 = 3609099.648$ $(699.5 - 225.58)^2 \times 23 = 224675.7764 \times 23 = 5167542.8472$ $(899.5 - 225.58)^2 \times 11 = 454162.9764 \times 11 = 4995792.7412$ $(1099.5 - 225.58)^2 \times 6 = 763650.1764 \times 6 = 4581901.0584$ $(1299.5 - 225.58)^2 \times 1 = 1153137.3764 \times 1 = 1153137.3764$

STEP 17

Sum the results from the previous step to find the total weighted squared differences.
$\text{Total Weighted Squared Differences} = 5498988.472 + 523899.654 + 3609099.648 + 5167542.8472 + 4995792.7412 + 4581901.0584 + 1153137.3764 = 30566361.7972$

STEP 18

Approximate the variance by dividing the total weighted squared differences by the total frequency minus one (n-1).
$\text{Variance} = \frac{30566361.7972}{533 - 1} = \frac{30566361.7972}{532} \approx 57473.23$

STEP 19

Approximate the standard deviation by taking the square root of the variance.
$\text{Standard Deviation} = \sqrt{57473.23} \approx 239.74$ Rounding to the nearest dollar: $\text{Standard Deviation} \approx 240$
The sample mean amount of savings is $\$$ 226, and the sample standard deviation is $\$$ 240.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the midpoint for each range of savings.
2. Use the midpoints and frequencies to approximate the mean savings.
3. Use the midpoints, frequencies, and the mean to approximate the standard deviation.

STEP 3

Calculate the midpoint for the first range $\$0 - \$199$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{0 + 199}{2} = 99.5$

STEP 4

Calculate the midpoint for the second range $\$200 - \$399$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{200 + 399}{2} = 299.5$

STEP 5

Calculate the midpoint for the third range $\$400 - \$599$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{400 + 599}{2} = 499.5$

STEP 6

Calculate the midpoint for the fourth range $\$600 - \$799$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{600 + 799}{2} = 699.5$

STEP 7

Calculate the midpoint for the fifth range $\$800 - \$999$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{800 + 999}{2} = 899.5$

STEP 8

Calculate the midpoint for the sixth range $\$1000 - \$1199$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{1000 + 1199}{2} = 1099.5$

STEP 9

Calculate the midpoint for the seventh range $\$1200 - \$1399$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{1200 + 1399}{2} = 1299.5$

STEP 10

Calculate the total number of respondents by summing the frequencies.
$\text{Total Frequency} = 348 + 96 + 48 + 23 + 11 + 6 + 1 = 533$

STEP 11

Calculate the weighted sum of the midpoints using the frequencies.
$\text{Weighted Sum} = 99.5 \times 348 + 299.5 \times 96 + 499.5 \times 48 + 699.5 \times 23 + 899.5 \times 11 + 1099.5 \times 6 + 1299.5 \times 1$

STEP 12

Perform the calculations for each term in the weighted sum.
$99.5 \times 348 = 34626$ $299.5 \times 96 = 28752$ $499.5 \times 48 = 23976$ $699.5 \times 23 = 16088.5$ $899.5 \times 11 = 9894.5$ $1099.5 \times 6 = 6597$ $1299.5 \times 1 = 1299.5$

STEP 13

Sum the results from the previous step to find the total weighted sum.
$\text{Total Weighted Sum} = 34626 + 28752 + 23976 + 16088.5 + 9894.5 + 6597 + 1299.5 = 120233.5$

STEP 14

Approximate the mean savings by dividing the total weighted sum by the total frequency.
$\text{Mean} = \frac{120233.5}{533} \approx 225.58$ Rounding to the nearest dollar: $\text{Mean} \approx 226$

STEP 15

STEP 16

STEP 17

Sum the results from the previous step to find the total weighted squared differences.
$\text{Total Weighted Squared Differences} = 5498988.472 + 523899.654 + 3609099.648 + 5167542.8472 + 4995792.7412 + 4581901.0584 + 1153137.3764 = 30566361.7972$

STEP 18

Approximate the variance by dividing the total weighted squared differences by the total frequency minus one (n-1).
$\text{Variance} = \frac{30566361.7972}{533 - 1} = \frac{30566361.7972}{532} \approx 57473.23$

STEP 19

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Studdy Solution

STEP 1

STEP 2

1. Calculate the midpoint for each range of savings.2. Use the midpoints and frequencies to approximate the mean savings.3. Use the midpoints, frequencies, and the mean to approximate the standard deviation.

STEP 3

Calculate the midpoint for the first range $0−$199\$0 - \$199$0−$199.The midpoint is calculated as follows: Midpoint=0+1992=99.5 \text{Midpoint} = \frac{0 + 199}{2} = 99.5 Midpoint=20+199​=99.5

STEP 4

Calculate the midpoint for the second range $200−$399\$200 - \$399$200−$399.The midpoint is calculated as follows: Midpoint=200+3992=299.5 \text{Midpoint} = \frac{200 + 399}{2} = 299.5 Midpoint=2200+399​=299.5

STEP 5

Calculate the midpoint for the third range $400−$599\$400 - \$599$400−$599.The midpoint is calculated as follows: Midpoint=400+5992=499.5 \text{Midpoint} = \frac{400 + 599}{2} = 499.5 Midpoint=2400+599​=499.5

STEP 6

Calculate the midpoint for the fourth range $600−$799\$600 - \$799$600−$799.The midpoint is calculated as follows: Midpoint=600+7992=699.5 \text{Midpoint} = \frac{600 + 799}{2} = 699.5 Midpoint=2600+799​=699.5

STEP 7

Calculate the midpoint for the fifth range $800−$999\$800 - \$999$800−$999.The midpoint is calculated as follows: Midpoint=800+9992=899.5 \text{Midpoint} = \frac{800 + 999}{2} = 899.5 Midpoint=2800+999​=899.5

STEP 8

Calculate the midpoint for the sixth range $1000−$1199\$1000 - \$1199$1000−$1199.The midpoint is calculated as follows: Midpoint=1000+11992=1099.5 \text{Midpoint} = \frac{1000 + 1199}{2} = 1099.5 Midpoint=21000+1199​=1099.5

STEP 9

Calculate the midpoint for the seventh range $1200−$1399\$1200 - \$1399$1200−$1399.The midpoint is calculated as follows: Midpoint=1200+13992=1299.5 \text{Midpoint} = \frac{1200 + 1399}{2} = 1299.5 Midpoint=21200+1399​=1299.5

STEP 10

Calculate the total number of respondents by summing the frequencies.Total Frequency=348+96+48+23+11+6+1=533 \text{Total Frequency} = 348 + 96 + 48 + 23 + 11 + 6 + 1 = 533 Total Frequency=348+96+48+23+11+6+1=533

STEP 11

STEP 12

STEP 13

STEP 14

Approximate the mean savings by dividing the total weighted sum by the total frequency.Mean=120233.5533≈225.58 \text{Mean} = \frac{120233.5}{533} \approx 225.58 Mean=533120233.5​≈225.58 Rounding to the nearest dollar: Mean≈226 \text{Mean} \approx 226 Mean≈226

STEP 15

STEP 16

STEP 17

STEP 18

STEP 19

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1. Calculate the midpoint for each range of savings.
2. Use the midpoints and frequencies to approximate the mean savings.
3. Use the midpoints, frequencies, and the mean to approximate the standard deviation.

Calculate the midpoint for the first range $\$0 - \$199$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{0 + 199}{2} = 99.5$

Calculate the midpoint for the second range $\$200 - \$399$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{200 + 399}{2} = 299.5$

Calculate the midpoint for the third range $\$400 - \$599$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{400 + 599}{2} = 499.5$

Calculate the midpoint for the fourth range $\$600 - \$799$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{600 + 799}{2} = 699.5$

Calculate the midpoint for the fifth range $\$800 - \$999$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{800 + 999}{2} = 899.5$

Calculate the midpoint for the sixth range $\$1000 - \$1199$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{1000 + 1199}{2} = 1099.5$

Calculate the midpoint for the seventh range $\$1200 - \$1399$ .
The midpoint is calculated as follows: $\text{Midpoint} = \frac{1200 + 1399}{2} = 1299.5$

Calculate the total number of respondents by summing the frequencies.
$\text{Total Frequency} = 348 + 96 + 48 + 23 + 11 + 6 + 1 = 533$

Approximate the mean savings by dividing the total weighted sum by the total frequency.
$\text{Mean} = \frac{120233.5}{533} \approx 225.58$ Rounding to the nearest dollar: $\text{Mean} \approx 226$