Questions: Recently, 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. The sample mean amount of savings is 225 (Round to the nearest dollar as needed) The sample standard deviation is . (Round to the nearest dollar as needed.)

Solution Steps

To approximate the mean and standard deviation of the amount of savings, we can use the frequency distribution data. The mean is already given as $225. To find the standard deviation, we will use the formula for the sample standard deviation, which involves calculating the squared differences from the mean, multiplying by the frequencies, summing them up, dividing by the total number of samples minus one, and then taking the square root.

The total number of samples \( n \) is the sum of the frequencies: \[ n = 5 + 10 + 15 + 20 + 10 + 5 = 65 \]

The sum of squared differences from the mean is given by: \[ \sum (f_i \cdot (x_i - \text{mean})^2) = 1215625 \]

The sample variance \( s^2 \) is calculated using the formula: \[ s^2 = \frac{\sum (f_i \cdot (x_i - \text{mean})^2)}{n - 1} \] Substituting the values: \[ s^2 = \frac{1215625}{65 - 1} = \frac{1215625}{64} = 18994.140625 \]

The sample standard deviation \( s \) is the square root of the sample variance: \[ s = \sqrt{18994.140625} \approx 137.8 \] Rounding to the nearest dollar: \[ s \approx 138 \]

Final Answer