A survey of 52 supermarkets yielded the following relative frequency table, where X is the number of checkout lanes at a randomly chosen supermarket.
$$
\begin{array}{c|cccccccccc}
X & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline
P(X=x) & 0.01 & 0.04 & 0.04 & 0.08 & 0.10 & 0.15 & 0.30 & 0.15 & 0.08 & 0.05
\end{array}
$$
(a) Compute µ = E(X). HINT [See Example 3.]
E(X) =
Interpret the result.
- This was the most frequently observed number of checkout lanes in surveyed supermarkets.
- There were at least this many checkout lanes in each supermarket that was surveyed.
- There were, on average, this many checkout lanes in a supermarket that was surveyed.
(b) Find P(X < μ) or P(X > μ).
P(x < μ) =
P(x > μ) =
The following table shows the approximate numbers of school goers in the United States (residents who attended some educational institution) in 1998, broken down by age group.
$$
\begin{array}{c|cccccc}
\text{Age} & 3-6.9 & 7-12.9 & 13-16.9 & 17-22.9 & 23-26.9 & 27-42.9 \\ \hline
\text{Population} & 10 & 20 & 18 & 13 & 3 & 5 \\
\text{(millions)} & & & & & &
\end{array}
$$
Use the rounded midpoints of the given measurement classes to compute the probability P distribution of the age X of a school goer. (Round probabilities to four decimal places.)
$$
\begin{array}{c|cccccc}
\text{Age} & 5 & 10 & 15 & 20 & 25 & 35 \\ \hline
P(X=x) & & & & & &
\end{array}
$$
Compute the expected value of X, E(X). (Round your answer to one decimal place.)
E(X) =
What information does the expected value give about residents enrolled in schools? (Round your answer to one decimal place.)
In 1998, the average age of a school goer was years old.
The following table shows the distribution of household incomes in 2010 for a sample of 1,000 households in a country with incomes up to $100,000.
$$
\begin{array}{c|cccccc}
\text{Income} & 0- & 20,000- & 40,000- & 60,000- & 80,000- & \\
\text{Bracket (\$)} & 19,999 & 39,999 & 59,999 & 79,999 & 99,999 & \\ \hline
\text{Households} & 230 & 280 & 200 & 170 & 120 &
\end{array}
$$
Use this information to estimate, to the nearest $1,000, the average household income for such households. HINT [See Example 6.]
$
Compute the (sample) variance and standard deviation of the data sample. (Round your answers to two decimal places.)
4, -4.4, 4.5, -0.5, -0.5
variance
standard deviation
Calculate the standard deviation of X for the probability distribution. (Round your answer to two decimal places.)
$$
\begin{array}{c|cccccc}
X & -5 & -1 & 0 & 2 & 5 & 10 \\ \hline
P(X=x) & 0.1 & 0.2 & 0.3 & 0.1 & 0.3 & 0
\end{array}
$$
Calculate the standard deviation o of X for the probability distribution. (Round your answer to two decimal places.)
$$
\begin{array}{c|cccccc}
X & -20 & -10 & 0 & 10 & 20 & 30 \\ \hline
P(X=x) & 0.3 & 0.1 & 0.1 & 0.3 & 0 & 0.2
\end{array}
$$
Calculate the expected value, the variance, and the standard deviation of the given random variable X. (Round all answers to two decimal places.)
X is the number of red marbles that Suzan has in her hand after she selects four marbles from a bag containing four red marbles and two green ones.
expected value
variance
standard deviation
Your company, Sonic Video, Inc., has conducted research that shows the following probability distribution, where X is the number of video arcades in a randomly chosen city with more than 500,000 inhabitants.
$$
\begin{array}{c|cccccccccc}
X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
P(X=x) & 0.04 & 0.12 & 0.40 & 0.20 & 0.15 & 0.03 & 0.02 & 0.02 & 0.01 & 0.01
\end{array}
$$
(a) Compute µ = E(X). HINT [See Example 3.]
E(X) =
Interpret the result.
- There are at most this many video arcades in each city with more than 500,000 inhabitants.
- There are, on average, this many video arcades in a city with more than 500,000 inhabitants.
- This is the most frequently observed number of video arcades in cities with more than 500,000 inhabitants.
- There are at least this many video arcades in each city with more than 500,000 inhabitants.
(b) Find P(X < μ) or P(X > μ).
P(x < μ) =
P(x > μ) =
Popularity Ratings In your bid to be elected class representative, you have your election committee survey five randomly chosen students in your class and ask them to rank you on a scale of 0-10. Your rankings are 1, 2, 5, 2, 5.
(a) Find the sample mean and standard deviation. (Round your answers to two decimal places.)
sample mean
standard deviation
(b) Assuming the sample mean and standard deviation are indicative of the class as a whole, in what range does the empirical rule predict that approximately 68% of the class will rank you?
Your candidacy for elected class representative is being opposed by Slick Sally. Your election committee has surveyed six of the students in your class and had them rank Sally on a scale of 0-10. The rankings were 2, 8, 7, 10, 3, 8.
(a) Find the sample mean and standard deviation. HINT [See Example 1 and Quick Examples 1 and 2.) (Round your answers to two decimal places.)
sample mean
standard deviation
(b) Assuming that the sample mean and standard deviation are indicative of the class as a whole, in what range does the empirical rule predict that approximately 95% of the class will rank Sally? (Include only practical values in your interval. Round your answers to two decimal places.)
Following is a sample of unemployment rates (in percentage points) in a country sampled from the period 1990-2004. (Round your answers to two decimal places.)
4.1, 4.7, 5.1, 5.5, 4.9
(a) Compute the mean and standard deviation of the given sample.
mean
standard deviation
(b) Fill in the blanks.
percentage points
percentage points
Assuming that the distribution of unemployment rates in the population is symmetric and bell shaped, 95% of the time, the unemployment rate is between and percentage points.
Population Age The following chart shows the ages of 250 randomly selected residents of a country.
Frequency
120
100
80
70
67
60
54
40
19
20
0
15-29
30-64
65-74 Age
Use the relative frequency distribution based on the (rounded) midpoints of the given measurement classes to obtain an estimate of the average age of a resident in the country. (Round the answer to one decimal place).
