help i do not understand how to solve this question: assume that blood pressure readings are normally distributed with a mean of 111 and standard deviation of 7. A researcher wishes to select people for a study but wants to exclude the top and bottom 10 percent. What would be the upper and lower readings to qualify people to participate in the study?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.10) and its Z score.
Insert ±Z into equation above and solve for score.
To find the upper and lower readings that would qualify people to participate in the study, you can use the concept of Z-scores. A Z-score represents the number of standard deviations a data point is away from the mean.
First, let's calculate the Z-score for the upper 10th percentile value:
Step 1: Convert the desired percentile (90th percentile) to a decimal. You subtract the percentile from 100, which gives us 10. So, 10th percentile is 0.10 in decimal form.
Step 2: Use the Z-score formula: Z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation.
Since we want the upper value, the Z-score will be positive.
So, we have: 0.10 = (X - 111) / 7.
Solving for X:
0.10 * 7 = X - 111
0.7 = X - 111
X = 111 + 0.7
X = 111.7
Therefore, the upper reading to qualify people to participate in the study is approximately 111.7.
Next, let's calculate the Z-score for the lower 10th percentile value:
Since we want the lower value, the Z-score will be negative.
So, we have: -0.10 = (X - 111) / 7.
Solving for X:
-0.10 * 7 = X - 111
-0.7 = X - 111
X = 111 - 0.7
X = 110.3
Therefore, the lower reading to qualify people to participate in the study is approximately 110.3.
In summary, to qualify people to participate in the study, their blood pressure readings should be between approximately 110.3 and 111.7.
To find the upper and lower readings to qualify people for the study, we need to determine the blood pressure values that correspond to the top and bottom 10% of the distribution.
1. Start by calculating the z-scores for the cutoff values using the standard normal distribution table or a statistical software:
- For the upper cutoff, find the z-score that corresponds to the top 10% probability. Since it is the upper tail, the z-score will be positive. By using the cumulative distribution function (CDF) of the standard normal distribution, we can find the z-score for which the area to the right is 10%.
- For the lower cutoff, find the z-score that corresponds to the bottom 10% probability. Since it is the lower tail, the z-score will be negative. Using the CDF, find the z-score for which the area to the left is 10%.
2. Once you have the z-scores, you can convert them back to actual blood pressure values using the formula:
z = (x - mean) / standard deviation
Rearrange the formula to solve for x:
x = (z * standard deviation) + mean
Let's calculate the upper and lower readings for this specific problem:
For the upper cutoff:
1. Using a standard normal distribution table (such as the Z-table), find the z-score corresponding to the probability of 0.10 (10% in the upper tail). The result is approximately 1.28.
2. Apply the formula:
x = (1.28 * 7) + 111
x ≈ 119.96 (approximately 120)
Therefore, the upper reading to qualify people for the study is approximately 120.
For the lower cutoff:
1. Using the same standard normal distribution table, find the z-score corresponding to the probability of 0.10 (10% in the lower tail). The result is approximately -1.28.
2. Apply the formula:
x = (-1.28 * 7) + 111
x ≈ 101.96 (approximately 102)
Therefore, the lower reading to qualify people for the study is approximately 102.
In summary, the upper cutoff for blood pressure readings to qualify for the study is approximately 120, and the lower cutoff is approximately 102.