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The 90% confidence interval that estimates the proportion of 8th graders that are involved in an after-school activity is (0.80853,0.93147).
The confidence interval for a population proportion for a given sample is given by the formula:
,
where p is the population proportion, Z is the Z-score value for the confidence interval, and n is the sample size.
In the question, we are given a random sample of 81 8th graders at a large suburban middle school. This implies that the sample size, n = 81. Also, we are told that they indicated 87% of them were involved with some type of after-school activity. This implies that the population proportion, p = 87% = 0.87.
We are asked to find the 90% confidence interval that estimates the proportion of them that are involved in an after-school activity.
Z-score (Z) corresponding to a 90% confidence interval is 1.645.
Thus, we can find the confidence interval as follows:
(0.87 - 1.645√{(0.87(1 - 0.87))/81},0.87 + 1.645√{(0.87(1 - 0.87))/81})
= (0.87 - 0.061469,0.87 + 0.061469)
= (0.80853,0.93147).
Thus, the 90% confidence interval that estimates the proportion of 8th graders that are involved in an after-school activity is (0.80853,0.93147).
Learn more about confidence intervals of population proportions at
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Answer:
For this case we want to find this probability:
And we can use the z score formula to see how many deviation we are within the mean and we got:
And for this case we know that within 3 deviation from the mean we have 99.7% of the values and that's the answer for this case.
Step-by-step explanation:
Previous concepts
The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".
Let X the random variable who represent the number of phone calls answered.
From the problem we have the mean and the standard deviation for the random variable X.
So we can assume
On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:
• The probability of obtain values within one deviation from the mean is 0.68
• The probability of obtain values within two deviation's from the mean is 0.95
• The probability of obtain values within three deviation's from the mean is 0.997
Solution to the problem
For this case we want to find this probability:
And we can use the z score formula to see how many deviation we are within the mean and we got:
And for this case we know that within 3 deviation from the mean we have 99.7% of the values and that's the answer for this case.