Use the binomial expansion equation to derive the following...

Use the binomial expansion equation to derive the following results for the case when v is much less than c. (a) (b) (c) [3] An astronaut travelling by Earth measures her pulse rate to be 80 beats per minute. The doctor at ground control records her pulse as 76 beats per minute. (a) Who measures the proper time interval, the astronaut or the doctor on Earth? (b) What is the time between beats measured by the astronaut? 0.75[s] (c) What is the time between beats measured by the doctor? 0.789[s] (d) What is the value of \gamma = 1/√[1-\beta 2]? 1.052 (e)What is the value of \beta =v/c for the astronaut's ship? 0.3132

Transcript

00:01 So, let us start with the concept which we are going to use here for this question.

00:06 We have to use z -score formula which will be equal to x -bar minus mu by sigma where x -bar is the sample mean, mu is the population mean, sigma is the standard deviation.

00:15 So, as per the question for the a part, we have to find the probability that a battery lasts more than 4 hours.

00:26 More than 4 hours.

00:28 Now more than 4 hour means to say 4 multiplied by 60 of minutes that will come out equals to 240 minutes.

00:36 So we need to calculate the area under the normal distribution curve to the right of 240 minutes.

00:44 So we just use first the z -scope formula which can be equals to x bar minus mu by sigma.

00:49 That one goes to be, see the x bar value is given 240 minus or mu is given to you.

00:58 260 by sigma is given to us that is 50.

01:01 So this will comes out equals to negative of 0 .4.

01:07 So we have to find the probability using the normal distribution table for the right tailed that is p of x greater than 2 to 40 minutes.

01:17 So that's going to goes to be p z greater than 2 negative to 0 .4.

01:22 So by using normal distribution table you will get this value 0 .6554.

01:29 So the probability for the lasting of 4 hours that means 240 minutes is about 0 .6554.

01:39 So this will be the required answer for first part of this question.

01:43 Now in b part we have to calculate the quartile of the the battery where need to find the values q1 and q2.

01:51 So first of all z is corresponding to the 75 percent means quartile 1 is the value of p where z greater than 2 z1 and that will becomes out to 0 .25 1 minus 0 .75 will become 0 .25.

02:10 So using the standard normal distribution for the first quartile, p value will comes to 0 .6745.

02:20 Now we have to find the quartile values...

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