A statistics practitioner determined that the mean and standard deviation of a data set were 120and 30, respectively. What can you say about the proportions of observations that lie between each of the following intervals?
a. 90 and 150
b. 60 and 180
c. 30 and 210
Z = (score-mean)/SD
Calculate the respective Z scores. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to these Z scores.
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To determine the proportions of observations that lie between each of the given intervals, we will use the concept of the standard normal distribution.
The standard normal distribution is a symmetrical distribution with a mean of 0 and a standard deviation of 1. By standardizing our given data set, we can convert it into a standard normal distribution, making it easier to calculate proportions.
To standardize a data set, we use the formula:
Z = (X - μ) / σ
where Z represents the standard score, X is the value of the observation, μ is the mean of the data set, and σ is the standard deviation.
Now, let's calculate the z-scores for the given intervals:
a. For the interval 90 and 150:
To calculate the z-scores, we use the formula:
Z1 = (90 - 120) / 30 = -1
Z2 = (150 - 120) / 30 = 1
To find the proportion of observations within this interval, we calculate the area under the standard normal curve between -1 and 1. This can be done using a standard normal distribution table or a statistical software.
b. For the interval 60 and 180:
Z1 = (60 - 120) / 30 = -2
Z2 = (180 - 120) / 30 = 2
Again, we calculate the area under the standard normal curve between -2 and 2 to find the proportion of observations within this interval.
c. For the interval 30 and 210:
Z1 = (30 - 120) / 30 = -3.33
Z2 = (210 - 120) / 30 = 3
Once more, we calculate the area under the standard normal curve between -3.33 and 3 to find the proportion of observations within this interval.
By referring to a standard normal distribution table or using statistical software, you can find the proportions of observations that lie between each of the given intervals.