6. The daily dinner bills in a local restaurant are normally distributed with a

Question

6. The daily dinner bills in a local restaurant are normally distributed with a mean of S30 and a standard deviation of S6. (Round to 4 decimal places) [20 points a. Define the random variable in words a eule or functon that assigns a numencal value to the 0 b. What is the probability that a randomly selected bill will be at least $40.50? c. What percentage of the bills will be less than $17.90? d. What percentage of the bills are between S23.00 and $35.00? e. What is the expected amount of the daily dinner bill? f. What is the variance for the amount of the daily bill? g- What are the minimum and maximum of the middle 95% of the bills? If twenty four of one day's bills had a value of at least $43.06, how many bills did the restaurant collect on that day? h.

Explanation / Answer

the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd ~ N(0,1)
mean ( u ) = 30
standard Deviation ( sd )= 6
b.
P(X < 40.5) = (40.5-30)/6
= 10.5/6= 1.75
= P ( Z <1.75) From Standard Normal Table
= 0.9599
P(X > = 40.5) = (1 - P(X < 40.5)
= 1 - 0.9599 = 0.0401
c.
P(X < 17.9) = (17.9-30)/6
= -12.1/6= -2.0167
= P ( Z <-2.0167) From Standard Normal Table
= 0.0219
d.
BETWEEN THEM
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 23) = (23-30)/6
= -7/6 = -1.1667
= P ( Z <-1.1667) From Standard Normal Table
= 0.1217
P(X < 35) = (35-30)/6
= 5/6 = 0.8333
= P ( Z <0.8333) From Standard Normal Table
= 0.7977
P(23 < X < 35) = 0.7977-0.1217 = 0.676
e.
expcted amount of mean = $30
f.
variance = sd^2 = 6^2 = 36
g.
LOWER/BELOW
P ( Z < x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is -1.959964
P( x-u/s.d < x - 30/6 ) = 0.025
That is, ( x - 30/6 ) = -1.959964
--> x = -1.959964 * 6 + 30 = 18.240
UPPER/TOP
P ( Z > x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is 1.959964
P( x-u / (s.d) > x - 30/6) = 0.025
That is, ( x - 30/6) = 1.959964
--> x = 1.959964 * 6+30 = 41.760

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