The 200 students in a school band will attend and awards dinner. A random survey of 25 of these students was conducted to determ
1 answer:
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Answer:
a)
b) We need a sample of at least 282 young men.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by
This Zscore is how many standard deviations the value of the measure X is from the mean. After finding the Z-score, we look at the z-score table and find the p-value 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. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
(a) What standard deviation must x have so that 99.7% of allsamples give an x within one-half inch of μ?
To solve this problem, we use the 68-95-99.7 rule. This rule states that:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviations of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we want 99.7% of all samples give X within one-half inch of . So
must have
and
must have
.
So
(b) How large an SRS do you need to reduce the standard deviationof x to the value you found in part (a)?
You know from government data that heights of young men are approximately Normal with standard deviation about 2.8 inches. This means that
The standard deviation of a sample of n young man is given by the following formula
We want to have
We need a sample of at least 282 young men.
Answer:
probability that all the 15 students selected are girls
Step-by-step explanation:
The selection is from a sample without replacement, so we use the hypergeometric distribution to solve this question.
Hypergeometric distribution:
The probability of x sucesses is given by the following formula:
In which:
x is the number of sucesses.
N is the size of the population.
n is the size of the sample.
k is the total number of desired outcomes.
Combinations formula:
is the number of different combinations of x objects from a set of n elements, given by the following formula.
All girls from the first group:
20 students, so
10 girls, so
5 students will be selected, so
We want all of them to be girls, so we find P(X = 5).
All girls from the second group:
20 students, so
5 girls, so
5 students will be selected, so
We want all of them to be girls, so we find P(X = 5).
All girls from the third group:
20 students, so
8 girls, so
5 students will be selected, so
We want all of them to be girls, so we find P(X = 5).
All 15 students are girls:
Groups are independent, so we multiply the probabilities:
probability that all the 15 students selected are girls