Question

Suppose x is a normal random variable with μ = 35 and σ = 10. find p(55.5 < x < 69.7).

Answer 1

Answer:

$\boxed{\boxed{P(55.5 < X< 69.7)=0.01992}}$

Step-by-step explanation:

We know that,

$Z=\dfrac{X-\mu}{\sigma}$

where,

Z = z-score,

X = raw score,

μ = mean,

σ = standard deviation.

So,

$=P(55.5 < X< 69.7)$

$=P(55.5-35 < X-35< 69.7-35)$

$=P(\dfrac{55.5-35}{10}< \dfrac{X-35}{10}< \dfrac{69.7-35}{10})$

$=P(\dfrac{55.5-35}{10}< Z< \dfrac{69.7-35}{10})$

$=P(2.05< Z<3.47)$

$=P(Z<3.47)-P(Z<2.05)$

$=0.99974-0.97982\\\\=0.01992$

Answer 2

The probability of $P\left({55.5<X<69.7}\right)$ is $\boxed{0.01992}$.

Further Explanation:

The Z score of the standard normal distribution can be obtained as,

${\text{Z}}=\frac{{X-\mu}}{\sigma}$

Here, Z is the standard normal value, $\mu$ represents the mean, $\sigma$ represents the standard deviation.

Given:

The mean of test is $\mu =35$.

The standard deviation of the waiting time is $\sigma=10$.

Explanation:

The probability of mean lies between 55.5 to 69.7 can be calculated as follows,

$\begin{aligned}{\text{Probability}}&=P\left({55.5<X<69.7}\right)\\&=P\left({55.5-\mu <X-\mu <69.7-\mu }\right)\\&=P\left({\frac{{55.5 - \mu }}{\sigma }<\frac{{X-\mu }}{\sigma }<\frac{{69.7-\mu }}{\sigma}}\right)\\\end{aligned}$

Further solve the above equation to obtain the probability.

$\begin{aligned}{\text{Probability}}&=\left({\frac{{55.5-35}}{{10}}<Z<\frac{{69.7-35}}{{10}}}\right)\\&=\left({2.05<Z<3.47}\right)\\\end{aligned}$

The probability of $P\left({Z<3.47}\right)=0.99974$.

The probability $P\left({Z<2.05}\right)=0.97982$.

The probability can be calculated as follows.

$\begin{aligned}{\text{Probability}}&=P\left({Z<3.47}\right)-P\left({Z<2.05}\right)\\&=0.99974-0.97982\\\end{aligned}$

Hence, the probability of $P\left({55.5<X<69.7}\right)$ is $\boxed{0.01992}$.

Learn more:

1. Learn more about normal distribution https://brainly.com/question/12698949

2. Learn more about standard normal distribution https://brainly.com/question/13006989

3. Learn more about confidence interval of mean https://brainly.com/question/12986589

Answer details:

Grade: College

Subject: Statistics

Chapter: Confidence Interval

Keywords: Z-score, Z-value, binomial distribution, standard normal distribution, standard deviation, test, measure, probability, low score, mean, repeating, indicated, normal distribution, percentile, percentage, proportion, empirical rule.