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Answer :
The mean of the given data set is 3.5. The variance of Angela's daily calls is approximately 1.4102. The approximate mean for the SP500 price changes is 0.48.
Determining the Mean for the Given Data Set
The data set is: 9, 6, 2, 0, 2, 3, 5, 2, 1, 5.
The mean (average) is calculated by adding up all the values and then dividing by the number of values.
Mean ([tex]\bar{x}[/tex]) = [tex]\frac{9 + 6 + 2 + 0 + 2 + 3 + 5 + 2 + 1 + 5}{10}[/tex]
= [tex]\frac{35}{10}[/tex]
= 3.5
Computing the Variance for Angela's Daily Calls
First, we find the mean number of calls. We do this by multiplying each call value by its frequency, summing these products, and then dividing by the total number of days.
Mean ([tex]\bar{x}[/tex]) = [tex]\frac{ (4 \times 1) + (5 \times 3) + (6 \times 4) + (7 \times 4) + (8 \times 2) }{1 + 3 + 4 + 4 + 2}[/tex]
= [tex]\frac{ 4 + 15 + 24 + 28 + 16 }{14}[/tex]
= [tex]\frac{87}{14}[/tex]
= 6.2143 (rounded to 4 decimal places)
Next, we calculate the variance. Variance (s^2) is determined by taking the sum of the squared differences between each data point and the mean, multiplied by their respective frequencies, and then dividing by the number of observations minus one (since this is a sample).
Variance (s^2) = [tex]\frac{ \sum [f(x) \times (x - \bar{x})^2] }{N - 1}[/tex]
= [tex]\frac{ (1 \times (4 - 6.2143)^2) + (3 \times (5 - 6.2143)^2) + (4 \times (6 - 6.2143)^2) + (4 \times (7 - 6.2143)^2) + (2 \times (8 - 6.2143)^2) }{14 - 1}[/tex]
= [tex]\frac{ 1 \times 4.897 + 3 \times 1.475 + 4 \times 0.0457 + 4 \times 0.614 + 2 \times 3.186 }{13}[/tex]
= [tex]\frac{ 4.897 + 4.425 + 0.1828 + 2.456 + 6.372 }{13}[/tex]
= [tex]\frac{ 18.3328 }{13}[/tex]
= 1.4102 (rounded to 4 decimal places)
Calculating the Mean for SP500 Price Changes Using Grouped Data
To find the mean, we multiply each midpoint by its corresponding proportion and then sum these products.
Mean = [tex]\sum (x \times p(x))[/tex]
[tex]= (-0.80 \times 0.03) + (-0.40 \times 0.03) + (0.00 \times 0.20) + (0.40 \times 0.44) + (0.80 \times 0.16) + (1.20 \times 0.06) + (1.60 \times 0.05) + (2.00 \times 0.03)[/tex]
= -0.024 - 0.012 + 0 + 0.176 + 0.128 + 0.072 + 0.08 + 0.06
= 0.48
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Rewritten by : Barada
The mean of the given data set is 3.5, the variance is 5.056, and the approximate mean using the grouped data table is 0.428.
The average of a set of values is referred to as the mean. To calculate the mean of a dataset, add up all of the values and divide by the number of values.
We can calculate the mean of the given data set as follows:
Mean = (9 + 6 + 2 + 0 + 2 + 3 + 5 + 2 + 1 + 5)/10
Mean = 3.5
Variance:Variance is a measure of how much a set of data varies. In other words, variance tells us how far each value in the dataset is from the mean. The variance formula is as follows:
σ² = (Σ(x - μ)²) / N
Where σ² is the variance, Σ is the sum, x is each value in the data set, μ is the mean of the data set, and N is the number of values in the data set.
Using the above formula, we can calculate the variance for the given data as follows:
First, we need to calculate the mean of the data set.x f(x)
xf(x)4 1 45 3 15 6 4 24 7 4 28 8 2 16
Total 14 128
Mean (μ) = 128 / 14 = 9.14
σ² = (Σ(x - μ)²) / N
σ² = [1(4 - 9.14)² + 3(5 - 9.14)² + 4(6 - 9.14)² + 4(7 - 9.14)² + 2(8 - 9.14)²] / 14
σ² = 5.056
Approximating the Mean:
We can use the midpoint of each interval as an approximate value for the entire interval. We can calculate the approximate mean as follows:
Mean = Σ(midpoint × proportion of stocks)
Mean = (-0.80 × 0.03) + (-0.40 × 0.03) + (0.00 × 0.2) + (0.40 × 0.44) + (0.80 × 0.16) + (1.20 × 0.06) + (1.60 × 0.05) + (2.00 × 0.03)
Mean = 0.428
Therefore, the mean of the given data set is 3.5, the variance is 5.056, and the approximate mean using the grouped data table is 0.428.
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