A random variable X follows the uniform distribution with a lower limit of 730 and an upper limit of 830.
a. Calculate the mean and the standard deviation for the distribution. (Round intermediate calculation for standard deviation to 4 decimal places and final answer to 2 decimal places.)
b. What is the probability that X is less than 820? (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
a. Since X follows a uniform distribution, the mean is equal to the average of the lower and upper limits.
Mean = (Lower limit + Upper limit) / 2 = (730 + 830) / 2 = 780
The standard deviation for a uniform distribution is given by the formula:
Standard deviation = (Upper limit - Lower limit) / sqrt(12) = (830 - 730) / sqrt(12) ≈ 10.240
Therefore, the mean is 780 and the standard deviation is approximately 10.240.
b. To find the probability that X is less than 820, we need to find the area under the probability density function (PDF) curve from the lower limit (730) to 820.
The PDF for a uniform distribution is equal to 1 / (Upper limit - Lower limit) in the range between the lower and upper limits.
Therefore, the probability that X is less than 820 can be calculated as follows:
Probability(X < 820) = (820 - 730) / (830 - 730) = 90 / 100 = 0.9
So, the probability that X is less than 820 is 0.9.
a. To calculate the mean and standard deviation for a uniform distribution, we can use the following formulas:
Mean (μ) = (Lower Limit + Upper Limit) / 2
Standard Deviation (σ) = (Upper Limit - Lower Limit) / √12
Given that the lower limit (a) is 730 and the upper limit (b) is 830, we can plug these values into the formulas:
Mean (μ) = (730 + 830) / 2 = 780
Standard Deviation (σ) = (830 - 730) / √12 ≈ 29.466
Therefore, the mean is 780 and the standard deviation is approximately 29.47.
b. To find the probability that X is less than 820, we need to calculate the z-score for 820 and then find the corresponding cumulative probability using a standard normal distribution table or a calculator.
The formula for calculating the z-score is:
z = (x - μ) / σ
Plugging in the values:
z = (820 - 780) / 29.47 ≈ 1.36
Using a standard normal distribution table or calculator, we can find the cumulative probability associated with a z-score of 1.36.
The probability that X is less than 820 is approximately 0.9131 or 91.31%.
a. To calculate the mean (μ) and the standard deviation (σ) for a uniform distribution, we can use the following formulas:
Mean: μ = (lower limit + upper limit) / 2
Standard deviation: σ = (upper limit - lower limit) / √12
In this case:
lower limit = 730
upper limit = 830
Let's calculate the mean first:
μ = (730 + 830) / 2
= 1560 / 2
= 780
So, the mean of the distribution is 780.
Now, let's calculate the standard deviation:
σ = (830 - 730) / √12
= 100 / √12
To round the standard deviation to 4 decimal places, we need to calculate the value of √12:
√12 ≈ 3.4641
So, substituting the value of √12 in the standard deviation formula:
σ = 100 / 3.4641
To calculate the standard deviation rounded to 4 decimal places:
σ ≈ 28.8675
Therefore, the standard deviation of the distribution is approximately 28.8675.
b. To calculate the probability that X is less than 820, we need to find the area under the probability density function (PDF) curve for the uniform distribution.
The PDF for a uniform distribution is a straight horizontal line, with the probability density being equal over the entire range.
The probability that X is less than 820 can be calculated as follows:
P(X < 820) = (820 - lower limit) / (upper limit - lower limit)
Substituting the given values:
P(X < 820) = (820 - 730) / (830 - 730)
= 90 / 100
= 0.9
Therefore, the probability that X is less than 820 is 0.9, or 90%.