Respuesta :
Using the normal distribution, the probability is given as follows:
[tex]P(X \geq 10) = 0.0485[/tex].
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score measures how many standard deviations the measure is above or below the mean.
- Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
- The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with [tex]\mu = np, \sigma = \sqrt{np(1-p)}[/tex].
The parameters for the binomial distribution are:
p = 0.5, n = 13.
Hence the mean and the standard deviation are:
- [tex]\mu = np = 13 \times 0.5 = 6.5[/tex].
- [tex]\sigma = \sqrt{np(1-p)} = \sqrt{13 \times 0.5 \times 0.5} = 1.8028[/tex]
Using continuity correction, the desired probability is P(X > 9.5), which is one subtracted by the p-value of Z when X = 9.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{9.5 - 6.5}{1.8028}[/tex]
Z = 1.66
Z = 1.66 has a p-value of 0.9515.
1 - 0.9515 = 0.0485, then:
[tex]P(X \geq 10) = 0.0485[/tex].
More can be learned about the normal distribution at https://brainly.com/question/4079902
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