5.1 basics of probability distributions. compute the mean and standard deviation of a discrete random…

5.1 basics of probability distributions. compute the mean and standard deviation of a discrete random variable.\nthe following table shows the number of people that have either 0, 1, 2 or 3 cars registered in their name, from a sample of 270 people.\na) complete the table.\n| x (the number of cars) | number of people |\n| ---- | ---- |\n| 0 | 62 |\n| 1 | 80 |\n| 2 | 78 |\n| 3 | i |\nb) find the mean number of cars owned for this sample. give your answer to at least 2 decimal places.\nc) find the standard deviation of cars owned for this sample. give your answer to at least 2 decimal places.\nquestion help: written example message instructor post to forum

5.1 basics of probability distributions. compute the mean and standard deviation of a discrete random variable.\nthe following table shows the number of people that have either 0, 1, 2 or 3 cars registered in their name, from a sample of 270 people.\na) complete the table.\n| x (the number of cars) | number of people |\n| ---- | ---- |\n| 0 | 62 |\n| 1 | 80 |\n| 2 | 78 |\n| 3 | i |\nb) find the mean number of cars owned for this sample. give your answer to at least 2 decimal places.\nc) find the standard deviation of cars owned for this sample. give your answer to at least 2 decimal places.\nquestion help: written example message instructor post to forum

Answer

Explanation:

Step1: Calculate the number of people with 3 cars

The total number of people in the sample is 270. Let the number of people with 3 cars be $n$. Then $n=270-(62 + 80+78)=270 - 220 = 50$.

Step2: Calculate the mean $\mu$

The formula for the mean of a discrete - random variable is $\mu=\sum_{i}x_ip_i$. Here, $x$ is the number of cars and $p$ is the probability. The probabilities $p_i$ are calculated as $p_i=\frac{\text{Number of People}}{\text{Total Number of People}}$. [ \begin{align*} \mu&=\frac{0\times62 + 1\times80+2\times78 + 3\times50}{270}\ &=\frac{0 + 80+156+150}{270}\ &=\frac{386}{270}\approx1.43 \end{align*} ]

Step3: Calculate the variance $\sigma^{2}$

The formula for the variance of a discrete - random variable is $\sigma^{2}=\sum_{i}(x_i-\mu)^{2}p_i$. [ \begin{align*} \sigma^{2}&=\frac{(0 - 1.43)^{2}\times62+(1 - 1.43)^{2}\times80+(2 - 1.43)^{2}\times78+(3 - 1.43)^{2}\times50}{270}\ &=\frac{( - 1.43)^{2}\times62+( - 0.43)^{2}\times80+(0.57)^{2}\times78+(1.57)^{2}\times50}{270}\ &=\frac{2.0449\times62 + 0.1849\times80+0.3249\times78+2.4649\times50}{270}\ &=\frac{126.7838+14.792+25.3422 + 123.245}{270}\ &=\frac{290.163}{270}\approx1.07 \end{align*} ]

Step4: Calculate the standard deviation $\sigma$

The standard deviation is the square - root of the variance. So $\sigma=\sqrt{\sigma^{2}}=\sqrt{1.07}\approx1.03$

Answer:

a) 50 b) 1.43 c) 1.03