#4 Find the value of x. Round your answer to the nearest tenth.
1 answer:
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Answer:
For a mean oil change time of 20.51 minutes there would be a 10% chance of being at or below
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
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.
In this problem, we have that:
Treating this as a random sample, at what mean oil-change time would there be a 10% chance of being at or below?
This is the 10th percentile, which is X when Z has a pvalue of 0.1. So it is X when Z = -1.28.
By the Central Limit Theorem
For a mean oil change time of 20.51 minutes there would be a 10% chance of being at or below
Answer:
The proportion of the population that have a protein requirement less than 0.60 g P • kg-1 • d-1 is 0.239, that is, 239 persons for every 1000, or simply 23.9% of them.
Step-by-step explanation:
From the question, we have the following information:
- The distribution for protein requirement is <em>normally distributed</em>.
- The population mean for protein requirement for adults is
- The population standard deviation is
We have here that protein requirements in adults is normally distributed with defined parameters. The question is about <em>the proportion</em> <em>of the population</em> that has a requirement less than .
For answering this, we need to calculate a <em>z-score</em> to obtain the probability of the value <em>x </em>in this distribution using a <em>standard normal table</em> available on the Internet or on any statistics book.
<h3>z-score</h3>A z-score is expressed as
For the given parameters, we have:
With this value for <em>z</em> at hand, we need to consult a standard normal table to determine what the probability of this value is.
The value for z = -0.7142857 is telling us that the requirement for protein is below the population mean (negative sign indicates this). However, most standard normal tables give a probability that a statistic is less than z and for values greater than the mean (in other words, positive values). To overcome this, we need to take the complement of the probability given for z-score z = 0.7142857, that is, subtract from 1 this probability, which is possible because the normal distribution is <em>symmetrical</em>.
Tables have values for <em>z</em> with two decimal places, then, for z = 0.7142857, we need to rewrite it as z = 0.71. For this value, the <em>standard normal table</em> gives a value of P(z<0.71) = 0.76115.
Therefore, the cumulative probability for values less than x = 0.60 which corresponds to a z-score = -0.7142857 is approximately:
(rounding to three decimal places)
That is, the proportion of the population that have a protein requirement less than 0.60 g P • kg-1 • d-1 is
See the graph below. The shaded area is the region that represents the proportion asked in the question.
