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Answer:
The lower bound is, [tex]-z=-0.69[/tex] and the upper bound is [tex]z=0.69[/tex].
Step-by-step explanation:
Let the random variable X follows a normal distribution with mean μ and standard deviation σ.
The the random variable Z, defined as [tex]Z=\frac{X-\mu}{\sigma}[/tex] is standardized random variable also known as a standard normal random variable. The random variable [tex]Z\sim N(0, 1)[/tex].
The standard normal random variable has a symmetric distribution.
It is provided that [tex]P(-z\leq Z\leq z)=0.51[/tex].
Determine the upper and lower bound as follows:
[tex]P(-z\leq Z\leq z)=0.51\\P(Z\leq z)-P(Z\leq -z)=0.51\\P(Z\leq z)-[1-P(Z\leq z)]=0.51\\2P(Z\leq z)-1=0.51\\2P(Z\leq z)=1.51\\P(Z\leq z)=0.755[/tex]
Use a standard normal table to determine the value of z.
The value of z such that P (Z ≤ z) = 0.755 is 0.69.
The lower bound is, [tex]-z=-0.69[/tex] and the upper bound is [tex]z=0.69[/tex].
The lower bound is -0.69 and the upper bound is 0.69.
Given
The probability is 51% that the sample means will be between what two values, symmetrically distributed around the population mean.
Symmetrical distribution
Symmetrical distribution is a situation in which the values of variables occur at regular frequencies, and the mean, median, and mode occur at the same point.
Let the random variable x follows a normal distribution with mean μ and standard deviation σ.
The random variable is defined as the;
[tex]\rm Z=\dfrac{X-\mu}{\sigma}[/tex]
The upper and lower bound is given as;
[tex]\rm P(-z\leq Z\leq z)=0.51\\\\P(Z\leq z)-P(Z\leq -z)=0.51\\\\P(Z\leq z)-(1-P(Z\leq z))=0.51\\\\2p(Z\leq z)-1=0.51\\\\2p(Z\leq z)=0.51+1\\\\2p(Z\leq z)=1.51\\\\p(Z\leq z)=\dfrac{1.51}{2}\\\\p(Z\leq z)=0.69[/tex]
Hence, the lower bound is -0.69 and the upper bound is 0.69.
To know more about normal distribution click the link given below.
https://brainly.com/question/14560346
