Question

at a swim meet, philip noted the number of members on various swim teams.

x is the number of swimmers that a randomly chosen team has. what is the standard deviation of x?

round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the total number of teams

$$3 + 4+6 + 2+5=20$$

Step2: Calculate the probability distribution

For \(x = 32\), \(P(X = 32)=\frac{3}{20}=0.15\)
For \(x = 36\), \(P(X = 36)=\frac{4}{20}=0.2\)
For \(x = 38\), \(P(X = 38)=\frac{6}{20}=0.3\)
For \(x = 39\), \(P(X = 39)=\frac{2}{20}=0.1\)
For \(x = 40\), \(P(X = 40)=\frac{5}{20}=0.25\)

Step3: Calculate the expected value \(E(X)\)

$$E(X)=\sum_{i}x_{i}P(X = x_{i})=32\times0.15 + 36\times0.2+38\times0.3+39\times0.1+40\times0.25$$
$$=4.8+7.2 + 11.4+3.9+10$$
$$=37.3$$

Step4: Calculate \(E(X^{2})\)

$$E(X^{2})=\sum_{i}x_{i}^{2}P(X = x_{i})=32^{2}\times0.15+36^{2}\times0.2 + 38^{2}\times0.3+39^{2}\times0.1+40^{2}\times0.25$$
$$=153.6+259.2+433.2+152.1+400$$
$$=1408.1$$

Step5: Calculate the variance \(\text{Var}(X)\)

$$\text{Var}(X)=E(X^{2})-(E(X))^{2}=1408.1-(37.3)^{2}$$
$$=1408.1 - 1391.29$$
$$=16.81$$

Step6: Calculate the standard deviation \(\sigma(X)\)

$$\sigma(X)=\sqrt{\text{Var}(X)}=\sqrt{16.81}\approx 4.10$$

Answer:

\(4.10\)