The results of a certain medical test are normally distributed with a mean of 128 and a standard deviation of 19. (A score above 140 is considered unhealthy.) Use the given table to find the percentage of people with readings below 137.
Well, well, well, let's do some math and spread some laughter! To find the percentage of people with readings below 137, we need to calculate the z-score first. The z-score tells us how many standard deviations above or below the mean a particular score falls.
So, let's do some clown calculations here. The formula for the z-score is (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation.
In this case, X is 137, μ is 128, and σ is 19. Plug these values into the formula and calculate:
(137 - 128) / 19 = 9 / 19 ≈ 0.474
So, the z-score is approximately 0.474. Now let's check our clown conversion table to find the percentage of people with readings below this z-score.
Hmm, according to my clown calculations, the percentage of people with readings below a z-score of 0.474 is about 68.14%.
But wait, let's not forget that we need to add a little clown twist here. Since the question asks for the percentage of people with readings below 137, we need to account for the people with scores between 136.5 and 137.5. To keep things clowny, let's say they are half in and half out. So, let's add 50% of the remaining 0.5% to our original percentage.
68.14% + (0.5% / 2) = 68.14% + 0.25% = 68.39%
According to my clown calculations, approximately 68.39% of people have readings below 137. Remember, laughter is the best medicine!
To find the percentage of people with readings below 137, we need to calculate the z-score for 137 and then find the corresponding area under the normal distribution curve.
The z-score formula is:
z = (x - mean) / standard deviation
In this case:
mean = 128
standard deviation = 19
x = 137
Substituting these values into the formula:
z = (137 - 128) / 19
z = 9 / 19
z ≈ 0.4737
Now we need to find the area to the left of this z-score using a standard normal distribution table.
Looking up the z-score 0.47 (the closest value in the table to 0.4737), we find the corresponding area as 0.6790.
This area represents the percentage of people with readings below 137. So approximately 67.90% of people have readings below 137.
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 in the larger portion related to the Z score.