Respuesta :
Answer:
14.28% of individual adult females have weights between 75 kg and 83 kg.
92.82% of the sample means are between 75 kg and 83 kg.
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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.
In this problem, we have that:
Assume that weights of adult females are normally distributed with a mean of 79 kg and a standard deviation of 22 kg. This means that [tex]\mu = 79, \sigma = 22[/tex].
What percentage of individual adult females have weights between 75 kg and 83 kg?
This percentage is the pvalue of Z when [tex]X = 83[/tex] subtracted by the pvalue of Z when [tex]X = 75[/tex]. So:
X = 83
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{83 - 79}{22}[/tex]
[tex]Z = 0.18[/tex]
[tex]Z = 0.18[/tex] has a pvalue of 0.5714.
X = 75
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{75- 79}{22}[/tex]
[tex]Z = -0.18[/tex]
[tex]Z = -0.18[/tex] has a pvalue of 0.4286.
This means that 0.5714-0.4286 = 0.1428 = 14.28% of individual adult females have weights between 75 kg and 83 kg.
If samples of 100 adult females are randomly selected and the mean weight is computed for each sample, what percentage of the sample means are between 75 kg and 83 kg?
Now we use the Central Limit THeorem, when [tex]n = 100[/tex]. So [tex]s = \frac{22}{\sqrt{100}} = 2.2.[/tex]
X = 83
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{83 - 79}{2.2}[/tex]
[tex]Z = 1.8[/tex]
[tex]Z = 1.8[/tex] has a pvalue of 0.9641.
X = 75
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{75-79}{2.2}[/tex]
[tex]Z = -1.8[/tex]
[tex]Z = -1.8[/tex] has a pvalue of 0.0359.
This means that 0.9641-0.0359 = 0.9282 = 92.82% of the sample means are between 75 kg and 83 kg.
93.12% of the sample means are between 75 kg and 83 kg
Z score
The z score is used to determine by how many standard deviations the raw score is above or below the mean. It is given by:
z = (x - μ)/(σ/√n)
where x is raw score, σ is standard deviation and μ is mean and n is sample size
μ = 79, σ = 22, n = 100
For x > 75:
z = (75 - 79)/(22/√100) = -1.82
For x < 83:
z = (83 - 79)/(22/√100) = 1.82
P(-1.82< z < 1.82) = P(z < 1.82) - P(z < -1.82) = 0.9656 - 0.0344 = 0.9312
93.12% of the sample means are between 75 kg and 83 kg
Find out more on Z score at: https://brainly.com/question/25638875
