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.
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
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