Question

calculate the population mean, variance, and standard deviation of seven test scores. remember to round appropriately.
66 71 80 100 68 83 74
\\(\mu = \square\\)
\\(\sigma = \square\\)
\\(\sigma^2 = \square\\)

Explanation:

Step1: Calculate the population mean ($\mu$)

The formula for the population mean is $\mu = \frac{\sum_{i = 1}^{N} x_i}{N}$, where $x_i$ are the data points and $N$ is the number of data points.
First, sum the test scores: $66 + 71 + 80 + 100 + 68 + 83 + 74 = 66 + 71 = 137; 137 + 80 = 217; 217 + 100 = 317; 317 + 68 = 385; 385 + 83 = 468; 468 + 74 = 542$.
There are $N = 7$ data points. So, $\mu=\frac{542}{7}\approx77.43$ (rounded to two decimal places).

Step2: Calculate the population variance ($\sigma^2$)

The formula for population variance is $\sigma^2=\frac{\sum_{i = 1}^{N}(x_i - \mu)^2}{N}$.
Calculate each $(x_i - \mu)^2$:

• For $x_1 = 66$: $(66 - 77.43)^2=(-11.43)^2 = 130.6449$
• For $x_2 = 71$: $(71 - 77.43)^2=(-6.43)^2 = 41.3449$
• For $x_3 = 80$: $(80 - 77.43)^2=(2.57)^2 = 6.6049$
• For $x_4 = 100$: $(100 - 77.43)^2=(22.57)^2 = 509.4049$
• For $x_5 = 68$: $(68 - 77.43)^2=(-9.43)^2 = 88.9249$
• For $x_6 = 83$: $(83 - 77.43)^2=(5.57)^2 = 31.0249$
• For $x_7 = 74$: $(74 - 77.43)^2=(-3.43)^2 = 11.7649$

Sum these squared differences: $130.6449+41.3449 = 171.9898; 171.9898+6.6049 = 178.5947; 178.5947+509.4049 = 687.9996; 687.9996+88.9249 = 776.9245; 776.9245+31.0249 = 807.9494; 807.9494+11.7649 = 819.7143$.
Now, divide by $N = 7$: $\sigma^2=\frac{819.7143}{7}\approx117.10$ (rounded to two decimal places).

Step3: Calculate the population standard deviation ($\sigma$)

The formula for population standard deviation is $\sigma=\sqrt{\sigma^2}$.
So, $\sigma=\sqrt{117.10}\approx10.82$ (rounded to two decimal places).

Answer:

$\mu\approx\boldsymbol{77.43}$, $\sigma\approx\boldsymbol{10.82}$, $\sigma^2\approx\boldsymbol{117.10}$ (answers are rounded to two decimal places; depending on rounding rules, slight variations may occur)