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Answer:
63% of the values lies between 70 and 85.
Step-by-step explanation:
We are given that a normally distributed set of numbers has a mean of 75 and a standard deviation of 7.97.
<em>Let X = Set of numbers</em>
The z-score probability distribution for is given by;
Z = ~ N(0,1)
where, = mean value = 75
= standard deviation = 7.97
The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.
So, percentage of values that lies between 70 and 85 is given by = P(70 < X < 85) = P(X < 85) - P(X 70)
P(X < 85) = P( <
) = P(Z < 1.25) = 0.89435 {using z table}
P(X 70) = P(
) = P(Z
-0.63) = 1 - P(Z < 0.63)
= 1 - 0.73565 = 0.26435
<em>Now, in the z table the P(Z </em><em> x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 1.25 and x = 0.63 in the z table which has an area of 0.89435 and 0.73565 respectively.</em>
Therefore, P(70 < X < 85) = 0.89435 - 0.26435 = 0.63 or 63%
<em>Hence, 63% of the values lies between 70 and 85.</em>
Answer:
a. 0.38%
b. 266.75 days
Step-by-step explanation:
We have the following data, mean (m) 269 and standard deviation (sd) 15, therefore:
a. The first thing is to calculate the number z:
z (x) = (x - m) / sd
z (309) = (309-269) / 15 = 2.67
When looking in the normal distribution table (attached), we have that at this value of z, the probability is:
P (z> 2.67), that is to say we must look in the table -2.67 and this value corresponds to 0.0038, that is to say 0.38%
b. Find the z-value with a left tail of 44%, i.e. 0.44. We look in the table for this value and what value of z corresponds.
invNorm (0.44) = -0.15
Find the corresponding number of days:
x = z * sd + m
we replace
d = -0.15 * 15 + 269 = 266.75 days
