suppose we have a set of blood pressure set with a mean of 120 systolic and a sample standard deviation of 20 points assume a normal distribution of what two values should 95% of all systolic bp lie
95% = mean ± 1.96 SD
Suppose the distribution of player heights can be considered symmetric and mound-shaped with mean 72.5 inches and standard deviation 1.2 inches. Use the 68-95-99.7 rule to answer the following questions.
To determine the range within which 95% of the systolic blood pressure (BP) values will lie, we can use the concept of z-scores and the properties of the normal distribution.
1. Start by finding the critical z-value for the 95% confidence level. The standard normal distribution has a mean of 0 and a standard deviation of 1. We need to find the z-value that corresponds to an area of 0.95 in the tails of the distribution. This critical z-value can be found using statistical tables or a calculator. For a two-sided confidence interval, we need to find the two critical z-values.
2. Once we have the critical z-values, we can use them to calculate the corresponding systolic BP values.
The formula to convert a z-score to the corresponding observation from a normal distribution is:
Observation = Mean + (z-score * Standard Deviation)
3. Applying the formula, calculate the lower and upper systolic BP values.
Here are the steps summarized:
1. Find the critical z-values for a 95% confidence level.
2. Calculate the systolic BP values using the following formula:
Lower Value = Mean - (z-score * Standard Deviation)
Upper Value = Mean + (z-score * Standard Deviation)
3. Substitute the values into the formula using the given mean (120) and sample standard deviation (20) to find the range within which 95% of the systolic BP values will lie.
Let's calculate it:
1. For a 95% confidence level, we need to find the critical z-values for the two tails of the distribution. The critical z-value for a two-tailed 95% confidence level is approximately 1.96.
2. Using the formula:
Lower Value = 120 - (1.96 * 20)
Upper Value = 120 + (1.96 * 20)
Lower Value = 120 - 39.2 = 80.8
Upper Value = 120 + 39.2 = 159.2
Therefore, we can say that 95% of all systolic BP values will lie between 80.8 and 159.2.