Final answer:
To find the 2% cutoff in a standard normal distribution, look for the z-score that corresponds to 2% of the data above it, which is about 2.05. To calculate the value that is two standard deviations above any mean, use the mean plus two times the standard deviation. For example, with a mean of 50 and a standard deviation of 6, the value is 62.
Step-by-step explanation:
To find the location of the 2% cutoff for the standard normal distribution, we look for the z-score that leaves 2% in the right tail of the distribution. This z-score, often notated as z0.02, can be found using statistical tables or a calculator equipped with statistical functions. Using such tools, we find that z0.02 is approximately 2.05. This means that 2% of the data lies above a z-score of 2.05.
To calculate the value that is two standard deviations above the mean in any normal distribution, we need the mean (denoted as μ) and standard deviation (denoted as σ). The empirical rule states that approximately 95% of the data in a normally distributed data set lies within two standard deviations of the mean. Therefore, to calculate the value that is two standard deviations above the mean, we use the formula: x = μ + (2 * σ). For example, if we are given a data set with a mean of 50 and a standard deviation of 6, the value two standard deviations above the mean would be x = 50 + (2 * 6) = 50 + 12 = 62. Similarly, for a sample mean (denoted as x) of 90 with a standard deviation of 0.1, two standard deviations above the mean would be calculated as x = 90 + (2 * 0.1) = 90 + 0.2 = 90.2.