A restaurant manager tracks the number of people in every party to sit at a specific table every day for a week, and then compiles the results into a probability distribution as shown in the table: Number of People , X 1 2 3 4 6 Relative Frequency, P(X) 0.05 0.18 0.22 0.06 0.03
We notice that the probability distribution only includes values from 1 to 6, and we assume that these are the only possible number of people in a party at this restaurant.
To find the mean, we can use the formula:
mean = Σ(X*P(X))
where Σ represents the sum of the values of the expression that follows it. So, we have:
mean = (1*0.05) + (2*0.18) + (3*0.22) + (4*0.06) + (6*0.03)
mean = 0.05 + 0.36 + 0.66 + 0.24 + 0.18
mean = 1.49
So the mean number of people in a party at this restaurant is 1.49.
To find the standard deviation, we can use the formula:
standard deviation = sqrt(Σ[(X-mean)^2*P(X)])
where sqrt represents the square root and Σ represents the sum of the values of the expression that follows it. So, we have:
standard deviation = sqrt([(1-1.49)^2*0.05] + [(2-1.49)^2*0.18] + [(3-1.49)^2*0.22] + [(4-1.49)^2*0.06] + [(6-1.49)^2*0.03])
standard deviation = sqrt([0.0025*0.05] + [0.2401*0.18] + [0.1044*0.22] + [0.0796*0.06] + [16.81*0.03])
standard deviation = sqrt(0.000125 + 0.043218 + 0.022968 + 0.004776 + 0.05043)
standard deviation = sqrt(0.121517)
standard deviation = 0.3488
So the standard deviation of the number of people in a party at this restaurant is 0.3488.
To find the expected value (mean) and standard deviation of the probability distribution, you can follow these steps:
Step 1: Calculate the expected value (mean)
The expected value is the sum of each value multiplied by its corresponding probability.
Expected Value (mean) = Σ(X * P(X))
Expected Value = (1 * 0.05) + (2 * 0.18) + (3 * 0.22) + (4 * 0.06) + (6 * 0.03)
Calculating this expression will give you the expected value (mean) of the probability distribution.
Step 2: Calculate the variance
The variance is the sum of the squared differences between each value and the expected value, multiplied by its corresponding probability.
Variance = Σ((X - Expected Value)^2 * P(X))
Variance = ((1 - Expected Value)^2 * 0.05) + ((2 - Expected Value)^2 * 0.18) + ((3 - Expected Value)^2 * 0.22) + ((4 - Expected Value)^2 * 0.06) + ((6 - Expected Value)^2 * 0.03)
Calculating this expression will give you the variance of the probability distribution.
Step 3: Calculate the standard deviation
The standard deviation is the square root of the variance.
Standard Deviation = √Variance
Calculating this expression will give you the standard deviation of the probability distribution.
Please note that you'll need to substitute the actual values for "Expected Value" in Step 2 to calculate the variance and Step 3 to calculate the standard deviation.
To calculate the expected value or average number of people in a party for the given probability distribution, you need to multiply each value (X) by its corresponding probability (P(X)), and then sum up the results.
Let's do the calculations step by step:
1. Multiply each value (X) by its corresponding probability (P(X)):
- For X = 1, the calculation is: 1 * 0.05 = 0.05
- For X = 2, the calculation is: 2 * 0.18 = 0.36
- For X = 3, the calculation is: 3 * 0.22 = 0.66
- For X = 4, the calculation is: 4 * 0.06 = 0.24
- For X = 6, the calculation is: 6 * 0.03 = 0.18
2. Sum up all the results from step 1:
0.05 + 0.36 + 0.66 + 0.24 + 0.18 = 1.49
The expected value or average number of people in a party for this probability distribution is 1.49.
Remember that the expected value represents the long-term average outcome, if the experiment (in this case, the number of people in a party) is repeated many times.