Uncle Henry has been having trouble keeping his weight constant. In fact, during each week, his weight changes from the beginning of the week to the end of the week by a random amount, uniformly distributed between -0.5 and 0.5 pounds. Assuming that his weight change during any given week is independent of his weight change during any other week, approximate the probability that at the end of 50 weeks Uncle Henry will have had a net change in weight of at least +3 pounds. You may want to refer to the standard normal table .
The answer can be calculated by Central Limit Theorem.
You need to find P(sn=>3) where sn= x1+x2+....+xn. So, to approximate this sum, we just need to calculate P(zn=>z) where zn=sn/(n^1/2)*standard deviation, and z= 3/2.0364. Finally, because we are dealing with the CDF of a normal random variable, we should calculate 1-P(zn<3/2.0364) by looking at standard normal table. The final answer is 0.0708.
To solve this problem, we can use the Central Limit Theorem to approximate the distribution of the net change in weight after 50 weeks. We can consider each week's weight change as a random variable with a uniform distribution between -0.5 and 0.5 pounds.
The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the shape of the original distribution.
In this case, we have 50 independent weight changes, each uniformly distributed between -0.5 and 0.5 pounds. The mean of each weight change is 0, and the standard deviation is (0.5 - (-0.5))/sqrt(12) = 1/√3 ≈ 0.577.
The net change in weight after 50 weeks will be the sum of these 50 weight changes. The mean of the net change will be 0, and the standard deviation will be √50 times the standard deviation of the individual weight changes, i.e., √50 * 0.577 ≈ 4.08.
To find the probability that the net change in weight is at least +3 pounds, we need to find the probability that a standard normal random variable is greater than (3 - 0)/4.08 ≈ 0.735.
Using a standard normal table (Z-table), we can look up the probability associated with the Z-score of 0.735. Checking the table, the probability is approximately 0.7673. Therefore, the approximate probability that at the end of 50 weeks Uncle Henry will have had a net change in weight of at least +3 pounds is 0.7673, or about 76.73%.
To solve this problem, we can use the concept of the Central Limit Theorem. The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables approaches a normal distribution.
In this case, we have 50 weeks, and each week's weight change is uniformly distributed between -0.5 and 0.5 pounds. The sum of these weekly weight changes over 50 weeks will also be normally distributed according to the Central Limit Theorem.
Let's define X as the net change in weight over 50 weeks. We want to find the probability that X is at least +3 pounds.
First, we need to calculate the mean and standard deviation of X. The mean of a uniformly distributed random variable is (a + b) / 2, where a is the lower bound and b is the upper bound. In this case, the mean of the weekly weight change is (0.5 + (-0.5)) / 2 = 0 pounds.
The standard deviation of a uniformly distributed random variable is (b - a) / sqrt(12), where a is the lower bound and b is the upper bound. In this case, the standard deviation of the weekly weight change is (0.5 - (-0.5)) / sqrt(12) ≈ 0.29 pounds.
The mean of X over 50 weeks will be 50 times the mean of the weekly weight change, which is 50 * 0 = 0 pounds. The standard deviation of X over 50 weeks will be the square root of 50 times the standard deviation of the weekly weight change, which is sqrt(50) * 0.29 ≈ 2.06 pounds.
Now, we can use the standard normal table to find the probability that a normally distributed random variable with mean 0 and standard deviation 2.06 is at least +3 pounds. We need to standardize the value of +3 pounds by subtracting the mean and dividing by the standard deviation.
Standardizing +3 pounds gives us (3 - 0) / 2.06 ≈ 1.46.
Looking up the value of 1.46 in the standard normal table, we find that the probability is approximately 0.9265.
Therefore, the approximate probability that at the end of 50 weeks Uncle Henry will have had a net change in weight of at least +3 pounds is 0.9265 or approximately 92.65%.