What is the range of this piecewise function
1 answer:
Answer as a compound inequality:
Answer in interval notation: [-4, 2)
=============================================
Explanation:
The range is the set of all possible y outputs of a function. When dealing with a graph like this, we just look at the highest and lowest points to determine which y values are possible.
The lowest point occurs when y = -4. We include this value. So far we have which is the same as
The upper ceiling for the y value is y = 2. We can't actually reach this value because of the open hole at (-3,2). So we say that
Combine and
to get the compound inequality
This says y is between -4 and 2, including -4 but excluding 2.
To convert this to interval notation, we write [-4, 2) where the square bracket says to include the endpoint and the curved parenthesis says to exclude the endpoint.
You might be interested in
Answer:
a) 0.0951
b) 0.8098
c) Between $24.75 and $27.25.
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:
(a)
What is the likelihood the sample mean is at least $27.00?
This is 1 subtracted by the pvalue of Z when X = 27. So
By the Central Limit Theorem
has a pvalue of 0.9049
1 - 0.9049 = 0.0951
(b)
What is the likelihood the sample mean is greater than $25.00 but less than $27.00?
This is the pvalue of Z when X = 27 subtracted by the pvalue of Z when X = 25. So
X = 27
has a pvalue of 0.9049
X = 25
has a pvalue of 0.0951
0.9049 - 0.0951 = 0.8098
c)Within what limits will 90 percent of the sample means occur?
50 - 90/2 = 5
50 + 90/2 = 95
Between the 5th and the 95th percentile.
5th percentile
X when Z has a pvalue of 0.05. So X when Z = -1.645
95th percentile
X when Z has a pvalue of 0.95. So X when Z = 1.645
Between $24.75 and $27.25.