STAT 2103 Lecture Notes - Lecture 5: Cumulative Distribution Function
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A salesperson makes four calls per year. A sample of 100 days given the following frequencies of sales volumes.
Number of Sales | Observed Frequency (days) |
0 | 30 |
1 | 32 |
2 | 25 |
3 | 10 |
4 | 3 |
TOTAL | 100 |
Records show sales are made to 30% of all sales calls. Assuming independent sales calls, the number of sales per day should follow a binomial distribution. The Binomial distribution is represented by:
For this exercise, assume that the population has a binomial probability distribution with n=4, p=0.30 and x= 0, 1, 2, 3 and 4.
(a) Compute the expected frequencies for x=0, 1, 2, 3 and 4 by using the binomial probability function. Combine categories if necessary to satisfy the requirement that the expected frequency is five or more for all categories.
(b) Use the goodness of fit test to determine whether the assumption of a binomial probability distribution should be rejected. Because no parameters of the Binomial probability distribution were estimated from the sample data, the degrees of freedom are k-1 .
The probability distribution of the random variable X represents the number of hits a baseball player obtained in a game for the 2012 baseball season
x |
P(x) |
0 |
0.167 |
1 |
0.3289 |
2 |
0.2801 |
3 |
0.149 |
4 |
0.0382 |
5 |
0.0368
|
The probability distribution was used along with statistical software to simulate 25 repetitions of the experiment (25 games). The number of hits was recorded. Approximate the mean and standard deviation of the random variable X based on the simulation. The simulation was repeated by performing 50 repetitions of the experiment. Approximate the mean and standard deviation of the random variable. Compare your results to the theoretical mean and standard deviation. What property is being illustrated?
a.Compute the theoretical mean of the random variable X for the given probability distribution.
Meu = ?
b. Compute the theoretical standard deviation of the random variable X for the given probability distribution.
Sigma =?
c.Approximate the mean of the random variable X based on the simulation for 25 games.
Xbar=?
d. Approximate the standard deviation of the random variable X based on the simulation for 25 games.
S=?
The probability distribution of the random variable X represents the number of hits a baseball player obtained in a game for the 2012 baseball season x P(x) 0 0.167 1 0.3289 2 0.2801 3 0.149 4 0.0382 5 0.0368 The probability distribution was used along with statistical software to simulate 25 repetitions of the experiment (25 games). The number of hits was recorded. Approximate the mean and standard deviation of the random variable X based on the simulation. The simulation was repeated by performing 50 repetitions of the experiment. Approximate the mean and standard deviation of the random variable. Compare your results to the theoretical mean and standard deviation. What property is being illustrated? a.Compute the theoretical mean of the random variable X for the given probability distribution. Meu = ? b. Compute the theoretical standard deviation of the random variable X for the given probability distribution. Sigma =? c.Approximate the mean of the random variable X based on the simulation for 25 games. Xbar=? d. Approximate the standard deviation of the random variable X based on the simulation for 25 games. S=?