The customer bills in a family restaurant are normally distributed. The bills mean is $28 and has a standard deviation of $6.
a. Identify what the random variable you are measuring is in this problem.
b. What is the probability that a randomly selected bill will be at least $39.10?
c. What percentage of the bills will be less than $16.90?
d. What are the minimum and maximum of the middle 95% of the bills?
e. If twelve of one day's bills had a value of at least $43.06, how many bills did the restaurant collect on that day?
a. The random variable being measured in this problem is the customer bills in the family restaurant.
b. To find the probability that a randomly selected bill will be at least $39.10, we can use the standard normal distribution.
First, we need to calculate the z-score for $39.10 using the formula: z = (x - μ) / σ
where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
In this case, x = $39.10, μ = $28, and σ = $6.
z = ($39.10 - $28) / $6 = 1.85
Next, we can use a z-table or calculator to find the probability corresponding to a z-score of 1.85.
The probability can be looked up as P(Z > 1.85) = 1 - P(Z < 1.85).
Using a z-table, this is approximately 0.0322.
So, the probability that a randomly selected bill will be at least $39.10 is approximately 0.0322, or 3.22%.
c. To find the percentage of bills that will be less than $16.90, we can again use the standard normal distribution.
First, we need to calculate the z-score for $16.90 using the formula: z = (x - μ) / σ
where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
In this case, x = $16.90, μ = $28, and σ = $6.
z = ($16.90 - $28) / $6 = -1.85
Next, we can use a z-table or calculator to find the probability corresponding to a z-score of -1.85.
The probability can be looked up as P(Z < -1.85).
Using a z-table, this is approximately 0.0322.
So, the percentage of the bills that will be less than $16.90 is approximately 0.0322, or 3.22%.
d. To find the minimum and maximum of the middle 95% of the bills, we need to calculate the z-scores for the corresponding percentile values.
The middle 95% corresponds to an area under the standard normal curve of 0.95.
Since this is a symmetric distribution, the z-scores for the middle 95% will be the values that cut off 2.5% from the lower and upper tails of the distribution.
Using a z-table or calculator, we find that the z-score for the lower 2.5% is approximately -1.96, and the z-score for the upper 2.5% is approximately 1.96.
Now we can calculate the actual values for these z-scores using the formula: x = μ + z * σ
For the lower bound: x = $28 + (-1.96) * $6 = $15.24
For the upper bound: x = $28 + (1.96) * $6 = $40.76
Therefore, the minimum and maximum of the middle 95% of the bills are $15.24 and $40.76, respectively.
e. If twelve of one day's bills had a value of at least $43.06, we can use this information to estimate how many bills the restaurant collected on that day.
Since we know that 12 bills had a value of at least $43.06, we can assume that the remaining bills had a value less than $43.06.
To estimate the total number of bills, we can calculate the proportion of bills that had a value less than $43.06 using the z-score.
First, we need to calculate the z-score for $43.06 using the formula: z = (x - μ) / σ
where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
In this case, x = $43.06, μ = $28, and σ = $6.
z = ($43.06 - $28) / $6 = 2.51
Next, we can use a z-table or calculator to find the proportion of bills that had a value less than $43.06, which corresponds to a z-score of 2.51.
Using a z-table, the proportion is approximately 0.9944.
Therefore, if 12 bills had a value of at least $43.06, the estimated total number of bills collected on that day can be calculated as:
Total number of bills = (12 / 0.9944)
Rounding to the nearest whole number, the estimated total number of bills collected on that day is 12.07 or approximately 12 bills.
a. The random variable being measured in this problem is the customer bills in the family restaurant.
b. To determine the probability that a randomly selected bill will be at least $39.10, you can use the Z-score formula. The Z-score measures the number of standard deviations a value is from the mean. In this case, the formula is:
Z = (X - μ) / σ
Where:
X = the value we want to find the probability for (in this case, $39.10)
μ = mean of the distribution ($28)
σ = standard deviation of the distribution ($6)
Substituting the values into the formula:
Z = (39.10 - 28) / 6
Z ≈ 1.85
Next, you can refer to a Z-table or use a statistical calculator to find the cumulative probability associated with a Z-score of 1.85. This will give you the probability that a randomly selected bill will be less than or equal to $39.10. To find the probability that the bill will be at least $39.10, subtract this probability from 1:
P(X ≥ $39.10) ≈ 1 - P(Z ≤ 1.85)
You can use the Z-table or a calculator to find the corresponding probability. Let's assume it is approximately 0.0322, or 3.22%.
Therefore, the probability that a randomly selected bill will be at least $39.10 is approximately 3.22%.
c. To determine the percentage of bills that will be less than $16.90, you can again use the Z-score formula and calculate the Z-score for $16.90. The formula remains the same:
Z = (X - μ) / σ
Substituting the values:
Z = (16.90 - 28) / 6
Z ≈ -1.85
To find the cumulative probability associated with a Z-score of -1.85, you can use the Z-table or a calculator. Let's assume it is approximately 0.0322, or 3.22%.
Therefore, the percentage of bills that will be less than $16.90 is approximately 3.22%.
d. To find the minimum and maximum of the middle 95% of the bills, you need to determine the Z-scores that correspond to the outer 2.5% of the distribution. This is because the middle 95% corresponds to the area between these two Z-scores.
Using the Z-table, you can find the Z-score that corresponds to an area of 0.025 to the left of it (to find the lower Z-score) and another Z-score that corresponds to an area of 0.025 to the right of it (to find the upper Z-score).
The Z-score corresponding to an area of 0.025 to the left is approximately -1.96.
The Z-score corresponding to an area of 0.025 to the right is approximately 1.96.
To find the minimum and maximum of the middle 95% of the bills, you can use the Z-score formula:
Minimum value = μ + (Z_lower * σ)
Maximum value = μ + (Z_upper * σ)
Substituting the values:
Minimum value = 28 + (-1.96 * 6)
Maximum value = 28 + (1.96 * 6)
Minimum value ≈ 15.24
Maximum value ≈ 40.76
Therefore, the minimum and maximum values of the middle 95% of the bills are approximately $15.24 and $40.76, respectively.
e. With the given information that twelve of one day's bills had a value of at least $43.06, you can use this data to estimate the total number of bills the restaurant collected on that day.
You can set up a proportion using the mean and standard deviation of the distribution. Assuming the distribution remains approximately normal:
(X - μ) / σ = (n - N) / √N
Where:
X = observed value ($43.06)
μ = mean of the distribution ($28)
σ = standard deviation of the distribution ($6)
n = observed number of bills (12)
N = total number of bills (to be determined)
Substituting the values:
(43.06 - 28) / 6 = (12 - N) / √N
Simplifying the equation:
15.06 / 6 = (12 - N) / √N
2.51 = (12 - N) / √N
Cross-multiply:
2.51√N = 12 - N
Square both sides to eliminate the square root:
6.3051N = 144 - 24N + N^2
Rearrange the equation to form a quadratic equation:
N^2 - 30.3051N + 144 = 0
Now, you can solve this quadratic equation to find the value of N, representing the total number of bills collected by the restaurant on that day. You can use the quadratic formula or factoring methods to solve it.
Once you find the value of N, you will have the estimated total number of bills collected by the restaurant on that day.
a. Customer bills
b. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
c. 95% = mean ± 1.96 SD
d. Calculate Z score and proportion.
proportion (x) = 12
Solve for x.