Answer :
The probabilities for the sample mean are given as follows:
a) 155 or more lbs: 0.8708 = 87.08%.
b) Between 150 and 153 lbs: 0.0467 = 4.67%.
How to use the normal distribution?
We first must use the z-score formula, as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
In which:
- X is the measure.
- [tex]\mu[/tex] is the population mean.
- [tex]\sigma[/tex] is the population standard deviation.
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, and can be positive(above the mean) or negative(below the mean).
The z-score table is used to obtain the p-value of the z-score, and it represents the percentile of the measure represented by X in the distribution.
By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation given by the equation presented as follows: [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
The parameters for this problem are given as follows:
[tex]\mu = 161, \sigma = \sqrt{729} = 27, n = 26, s = \frac{27}{\sqrt{26}} = 5.295[/tex]
The probability in item a is one subtracted by the p-value of Z when X = 155, hence:
Z = (155 - 161)/5.295
Z = -1.13
Z = -1.13 has a p-value of 0.1292.
Hence:
1 - 0.1292 = 0.8708 = 87.08%.
For item b, the probability is the p-value of Z when X = 153 subtracted by the p-value of Z when X = 150, hence:
Z = (153 - 161)/5.295
Z = -1.51
Z = -1.51 has a p-value of 0.0655.
Z = (150 - 161)/5.295
Z = -2.08
Z = -2.08 has a p-value of 0.0188.
0.0655 - 0.0188 = 0.0467 = 4.67%.
More can be learned about the normal distribution at https://brainly.com/question/25800303
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