Question

1. x represents the amount of snow (in inches) that fell in nome, alaska last winter

1. x represents the total number of die rolls required for an individual to roll a five

directions: use the frequency distribution to construct a probability distribution. then, find the mean and standard deviation of the probability distribution.

9 dogs the number of dogs per household in a small town

dogs	0	1	2	3	4	5

Explanation:

Step1: Calculate total number of households

First, we find the total number of households by summing up all the frequencies.
Total households \( N = 1491 + 425 + 168 + 48 + 29 + 14 \)
\( N = 1491+425 = 1916 \); \( 1916 + 168 = 2084 \); \( 2084+48 = 2132 \); \( 2132 + 29 = 2161 \); \( 2161+14 = 2175 \)

Step2: Construct probability distribution

The probability \( P(x) \) for each number of dogs \( x \) is given by \( P(x)=\frac{\text{Frequency of }x}{N} \)

• For \( x = 0 \): \( P(0)=\frac{1491}{2175}\approx0.6855 \)
• For \( x = 1 \): \( P(1)=\frac{425}{2175}\approx0.1954 \)
• For \( x = 2 \): \( P(2)=\frac{168}{2175}\approx0.0772 \)
• For \( x = 3 \): \( P(3)=\frac{48}{2175}\approx0.0221 \)
• For \( x = 4 \): \( P(4)=\frac{29}{2175}\approx0.0133 \)
• For \( x = 5 \): \( P(5)=\frac{14}{2175}\approx0.0064 \)

Step3: Calculate the mean (\(\mu\))

The formula for the mean of a probability distribution is \( \mu=\sum_{i}x_iP(x_i) \)
\[

$$\begin{align*} \mu&=0\times\frac{1491}{2175}+1\times\frac{425}{2175}+2\times\frac{168}{2175}+3\times\frac{48}{2175}+4\times\frac{29}{2175}+5\times\frac{14}{2175}\\ &=\frac{0 + 425+336 + 144+116 + 70}{2175}\\ &=\frac{425+336=761; 761 + 144=905; 905+116 = 1021; 1021+70 = 1091}{2175}\\ &=\frac{1091}{2175}\approx0.5016 \end{align*}$$

\]

Step4: Calculate the variance (\(\sigma^2\))

The formula for variance is \( \sigma^2=\sum_{i}(x_i - \mu)^2P(x_i) \)
\[

$$\begin{align*} &(0 - 0.5016)^2\times\frac{1491}{2175}+(1 - 0.5016)^2\times\frac{425}{2175}+(2 - 0.5016)^2\times\frac{168}{2175}+(3 - 0.5016)^2\times\frac{48}{2175}+(4 - 0.5016)^2\times\frac{29}{2175}+(5 - 0.5016)^2\times\frac{14}{2175}\\ &=(- 0.5016)^2\times\frac{1491}{2175}+(0.4984)^2\times\frac{425}{2175}+(1.4984)^2\times\frac{168}{2175}+(2.4984)^2\times\frac{48}{2175}+(3.4984)^2\times\frac{29}{2175}+(4.4984)^2\times\frac{14}{2175}\\ &=0.2516\times\frac{1491}{2175}+0.2484\times\frac{425}{2175}+2.2452\times\frac{168}{2175}+6.2420\times\frac{48}{2175}+12.2390\times\frac{29}{2175}+20.2366\times\frac{14}{2175}\\ \end{align*}$$

\]
Calculate each term:

• First term: \( 0.2516\times1491\approx375.1356 \); \( \frac{375.1356}{2175}\approx0.1725 \)
• Second term: \( 0.2484\times425\approx105.57 \); \( \frac{105.57}{2175}\approx0.0485 \)
• Third term: \( 2.2452\times168\approx377.1936 \); \( \frac{377.1936}{2175}\approx0.1734 \)
• Fourth term: \( 6.2420\times48\approx299.616 \); \( \frac{299.616}{2175}\approx0.1377 \)
• Fifth term: \( 12.2390\times29\approx354.931 \); \( \frac{354.931}{2175}\approx0.1632 \)
• Sixth term: \( 20.2366\times14\approx283.3124 \); \( \frac{283.3124}{2175}\approx0.1303 \)

Sum these terms: \( 0.1725 + 0.0485+0.1734 + 0.1377+0.1632 + 0.1303 = 0.8256 \)

Step5: Calculate the standard deviation (\(\sigma\))

Standard deviation is the square root of variance: \( \sigma=\sqrt{\sigma^2}=\sqrt{0.8256}\approx0.9086 \)

Answer:

• Probability Distribution:

Dogs (\(x\))	0	1	2	3	4	5

• Mean: \(\approx0.50\) (or more precisely \(0.5016\))
• Standard Deviation: \(\approx0.91\) (or more precisely \(0.9086\))