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stinkaq
17.05.2020 •
Mathematics
A company makes car batteries and claims 80% of its ABC batteries are good for 70 months or longer. Assume that this claim is true. Let p ˆ be the proportion in a sample of 100 such ABC batteries. What is the probability that this sample proportion is within 0.05 of the population proportion.
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Ответ:
78.88% probability that this sample proportion is within 0.05 of the population proportion
Step-by-step explanation:
We need to understand the normal probability distribution and the central limit theorem to solve this question.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure 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.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For proportion p in a sample of size n, we have that![\mu = p, s = \sqrt{\frac{\pi(1-\pi)}{n}}](/tpl/images/0653/8649/cf7a3.png)
In this question:
So
What is the probability that this sample proportion is within 0.05 of the population proportion.
This is the pvalue of Z when X = 0.8 + 0.05 = 0.85 subtracted by the pvalue of Z when X = 0.8 - 0.05 = 0.75.
X = 0.85
By the Central Limit Theorem
X = 0.75
0.8944 - 0.1056 = 0.7888
78.88% probability that this sample proportion is within 0.05 of the population proportion
Ответ:
Answer/Explanation:
A. Fom the dattached diagram below showing the number of flowers that have bloomed in the garden in the month of March, we can infer that the trend shows a strong positive correlation. If we attempt to draw a straight line to connect some of the points on the plot, we would realized that we're going to be having a line that slopes upward, and also points are either on the line or are at least not far from the line drawn. Having a correlation coefficient or r = 0.98, we can assume that value to be accurate because the trend shows a positive correlation that is closer to a correlation coefficient of 1.
B. A scenario that would show a casual relationship for flowers in the garden is the comparison of the numbers of flowers in the garden and the quantity, in kg of a nutrient-rich fertilizer.