Question

Assume the time spent (in days) waiting for a heart transplant for people ages 35-49 can be approximated by a normal distribution with mean of 203 days and a standard deviation of 25.7 days. what waiting time represents the 5th percentile? what waiting time represents the third quartile?

Answer 1

Answer:

5th percentile = 160.852 days

3rd quartile = 220.22 days

Step-by-step explanation:

Let X be the random variable representing the waiting time.

Here the mean(\mu) = 203 days

Standard deviation (\sigma) = 25.7 days

For calculating the 5th percentile, we need to find the z- score having area 0.05.

By Z-score table we have Z= -1.64

Now, Using formula X=Z\sigma +\mu

X = (-1.64)*(25.7)+203

  = 160.852 days

Waiting time representing 5th percentile = 160.852 days.

For calculating third quartile, we know that third quartile represents the area to the left of 75% = 0.75

Hence, we need to calculate the Z score when the area is 0.75. Using the table we get Z= 0.67

X = (0.67)*(25.7)+203

   = 220.22 (days)

Waiting time representing third quartile = 220.22 days

Answer 2

The waiting time represents the $5th$ percentile is $\boxed{160.73{\text{ }}\,{\text{days}}}$.

The waiting time represents the $3rd$ quartile is $\boxed{220.22{\text{ }}\,{\text{days}}}$.

Further Explanation:

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

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

Given:

The mean of test is $\boxed{203}$.

The standard deviation of the waiting time is $\boxed{25.7}$.

Explanation:

The value of z-score with fifth percentile can be obtained from the table is $1.6449$.

The z-score is $1.6449$.

The waiting time represents the $5th$ percentile can be obtained as,

$\begin{aligned} \left( { - 1.6449} \right) &= \frac{{{\text{X}} - 203}}{{25.7}} \hfill \\ \left( { - 1.6449} \right) \times \left( {25.7} \right) &= {\text{X}} - 203 \hfill \\ - 42.27 &= {\text{X}} - 203 \hfill \\ - 42.27 + 203 &= {\text{X}} \hfill \\ {\text{160}}{\text{.73}&= X}} \hfill \\ \end{aligned}$

The waiting time represents the $5th$ percentile is $\boxed{160.73{\text{ }}\,{\text{days}}}$.

The value of z-score with third quartile can be obtained from the table is $0.67$.

The z-score is $0.67$.

The waiting time represents the 3rd quartile can be obtained as,

$\begin{aligned} 0.67 &= \frac{{{\text{X}} - 203}}{{25.7}} \hfill \\ 0.67 \times \left( {25.7} \right) &= {\text{X}} - 203 \hfill \\ 17.19 + 203 &= {\text{X}} \hfill \\ {\text{220}}{\text{.22}} &= {\text{X}} \hfill \\ \end{aligned}$

The waiting time represents the $3rd$ quartile is $\boxed{220.22{\text{ }}\,{\text{days}}}$.

Learn more:

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

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

3. Learn more about confidence interval of mean 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, criminologist, test, measure, probability, low score, mean, repeating, indicated, normal distribution, percentile, percentage, undesirable behavior, proportion, empirical rule.