Respuesta :
Answer:
0.3960
Step-by-step explanation:
X - no of questions answered correct is binomial since each question is independent of the other and there are two outcomes
p = Prob of any one question right = 0.2 (1/5)
n = 44
Normal approximation would be mean = np = 8.8 and Variance = npq = 7.04
Std dev = [tex]\sqrt{7.04} \\=2.65335[/tex]
X is N(8.8,2.6534)
Or Z = [tex]\frac{x-8.8}{2.6534}[/tex]
Since we approximate discrete to continuous continuity correction is to be applied
[tex]x\geq 10[/tex] means x≥9.5
P(X≥9.5) = 0.3960
Using the normal approximation to the binomial, it is found that there is a 0.3959 = 39.59% probability this student obtains a score greater than or equal to 10.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It 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, 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].
In this problem:
- There are 44 questions, hence [tex]n = 44[/tex].
- Each question has five options, one of which are corrected, and since they are answered by guess-work, [tex]p = \frac{1}{5} = 0.2[/tex].
The mean and the standard deviation for the approximation are:
[tex]\mu = np = 44(0.2) = 8.8[/tex]
[tex]\sigma = \sqrt{np(1 - p)} = \sqrt{44(0.2)(0.8)} = 2.6533[/tex]
The probability that the score is of at least 10, using continuity correction, is [tex]P(X \geq 10 - 0.5) = P(X \geq 9.5)[/tex], which is 1 subtracted by the p-value of Z = 9.5. Hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{9.5 - 8.8}{2.6533}[/tex]
[tex]Z = 0.264[/tex]
[tex]Z = 0.264[/tex] has a p-value of 0.6041.
1 - 0.6041 = 0.3959.
0.3959 = 39.59% probability this student obtains a score greater than or equal to 10.
A similar problem is given at https://brainly.com/question/24261244
