A square poster has a side length of 26 in. Drawn on the poster are four identical triangles. Each triangle has a base of 8 in.
2 answers:
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Answer:
a) The mean of a sampling distribution of is
. The standard deviation is
.
b) The standard normal z-score corresponding to a value of is
.
c) The standard normal z-score corresponding to a value of is
.
d) The probability .
e) The probability .
f) .
Step-by-step explanation:
We are dealing here with the concept of <em>a sampling distribution</em>, that is, the distribution of the sample means .
We know that for this kind of distribution we need, at least, that the sample size must be observations, to establish that:
In words, the distribution of the sample means follows, approximately, a <em>normal distribution</em> with mean, , and standard deviation (called <em>standard error</em>),
.
The number of observations is n = 64.
We need also to remember that the random variable Z follows a <em>standard normal distribution</em> with and
.
The variable Z is
[1]
With all this information, we can solve the questions.
Part a
The mean of a sampling distribution of is the population mean
or
.
The standard deviation is the population standard deviation divided by the root square of n, that is, the number of observations of the sample. Thus,
.
Part b
We are dealing here with a <em>random sample</em>. The z-score for the sampling distribution of is given by [1]. Then
Then, the <em>standard normal z-score</em> corresponding to a value of is
.
Part c
We can follow the same procedure as before. Then
As a result, the <em>standard normal z-score</em> corresponding to a value of is
.
Part d
Since we know from [1] that the random variable follows a <em>standard normal distribution</em>, we can consult the <em>cumulative standard normal table</em> for the corresponding already calculated. This table is available in Statistics textbooks and on the Internet. We can also use statistical packages and even spreadsheets or calculators to find this probability.
The corresponding value is Z = -2, that is, it is <em>two standard units</em> <em>below</em> the mean (because of the <em>negative</em> value). Then, consulting the mentioned table, the corresponding cumulative probability for Z = -2 is .
Therefore, the probability .
Part e
We can follow a similar way than the previous step.
For using the <em>cumulative standard normal table</em>, we can find this probability knowing that
Thus
Therefore, the probability .
Part f
This probability is and
.
For finding this, we need to subtract the cumulative probabilities for and
Using the previous <em>standardized values</em> for them, we have from <em>Part d</em>:
We know from <em>Part e</em> that
Therefore, .