The random variable x is normally distributed with mean =1,000 and standard deviation =100. Sketch and find each of the following probabilities: P(x<1,035)
To find the probability P(x < 1,035) for a normally distributed random variable x with a mean of 1,000 and a standard deviation of 100, we can use the standard normal distribution table or calculate it using the z-score formula.
Step 1: Calculate the z-score
The z-score is a measure of how many standard deviations an observation is from the mean. It can be calculated using the formula:
z = (x - μ) / σ
where:
z is the z-score,
x is the value we want to find the probability for (1,035 in this case),
μ is the mean of the distribution (1,000 in this case),
and σ is the standard deviation of the distribution (100 in this case).
Substituting the given values:
z = (1,035 - 1,000) / 100
z = 35 / 100
z = 0.35
Step 2: Find the probability using the standard normal distribution table
The standard normal distribution table provides the probabilities associated with different values of the z-score. It represents the area under the standard normal curve.
In this case, we want to find P(z < 0.35), which represents the probability of getting a z-score less than 0.35. Looking up the z-score 0.35 in the standard normal distribution table, we find the corresponding probability to be approximately 0.6368.
Step 3: Interpret the probability
P(x < 1,035) = P(z < 0.35) = 0.6368
Therefore, the probability of x being less than 1,035 is approximately 0.6368.