Question

compute the standard deviation of the data set. round to the nearest hundredth, if needed. 4.9, 4.9, 9.9, 9.9, 14.9 a. 3.74 c. 8.9 b. 4.18 d. 14

Explanation:

Step1: Calculate the mean ($\mu$) of the data set.

The data set is \( 4.9, 4.9, 9.9, 9.9, 14.9 \). The number of data points \( n = 5 \).
The mean \(\mu=\frac{4.9 + 4.9 + 9.9 + 9.9 + 14.9}{5}\)
First, sum the values: \(4.9+4.9 = 9.8\), \(9.8+9.9 = 19.7\), \(19.7+9.9 = 29.6\), \(29.6+14.9 = 44.5\)
Then, \(\mu=\frac{44.5}{5}=8.9\)

Step2: Calculate the squared differences from the mean.

For each data point \( x \), calculate \((x - \mu)^2\):

• For \( x = 4.9 \): \((4.9 - 8.9)^2=(-4)^2 = 16\) (and there are two such points, so total for these two: \(2\times16 = 32\))
• For \( x = 9.9 \): \((9.9 - 8.9)^2=(1)^2 = 1\) (and there are two such points, so total for these two: \(2\times1 = 2\))
• For \( x = 14.9 \): \((14.9 - 8.9)^2=(6)^2 = 36\) (one such point, so total: \(36\))

Step3: Calculate the variance ($\sigma^2$).

Variance is the average of the squared differences.
Sum of squared differences: \(32 + 2+ 36=70\)
Variance \(\sigma^2=\frac{70}{5}=14\)

Step4: Calculate the standard deviation ($\sigma$).

Standard deviation is the square root of the variance.
\(\sigma=\sqrt{14}\approx3.74\) (rounded to the nearest hundredth)

Answer:

a. 3.74