Final answer:
To determine the probability that x is greater than 30, calculate the z-score and then use standard normal distribution tables or a calculator. The z-score is 2, indicating that 30 is 2 standard deviations from the mean. Approximately 2.5% of values are expected to be greater than 30.
Step-by-step explanation:
If we are given a normally distributed random variable x with a mean (μ) of 22 and a variance (σ2) of 16, we can find the probability that x is greater than a specific value (such as 30) by first calculating the z-score for the value 30. The z-score is the number of standard deviations a data point is from the mean. Since the standard deviation (σ) is the square root of the variance, we have σ = 4.a
To calculate the z-score for 30, we use the formula z = (x - μ) / σ, which yields z = (30 - 22) / 4, so z = 2. This means that the value 30 is 2 standard deviations above the mean. Using the z-score we can then calculate the probability that x is greater than 30 using standard normal distribution tables or a calculator. This process of finding the probability that a normally distributed random variable falls within a certain range is a fundamental concept in statistics.
According to the Empirical Rule, approximately 95 percent of the data values lie within two standard deviations of the mean. Since a z-score of 2 corresponds to the value 30, and the value 30 is exactly two standard deviations above the mean, we can estimate that 97.5% of the values of x will be less than 30 (100% - the remaining 2.5% above). However, a more precise method is to consult the standard normal distribution table or use a calculator that gives us the area to the right of the z-score of 2 (the chance that x is greater than 30) which would typically yield a probability of around 2.5%.