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Solve question 1,2,3,4 Skill- theet

[7]

Exercise 15C Note that many of the rea...

May 12, 2024

Solve question 1,2,3,4

Solution by Steps

step 1

Define the total time taken for a meal to be served as the sum of two independent normally distributed variables: preparation time and cooking time

step 2

Calculate the mean of the total time by adding the means of the preparation time and cooking time:

12 + 14 = 26

minutes

step 3

Calculate the variance of the total time by adding the variances of the preparation time and cooking time:

3^2 + 3^2 = 18

step 4

Calculate the standard deviation of the total time by taking the square root of the variance:

\sqrt{18} = 4.2426

minutes

step 5

Convert the problem to a standard normal distribution problem by finding the Z-score for 30 minutes:

Z = \frac{30 - 26}{4.2426} = 0.9428

step 6

Look up the Z-score in the standard normal distribution table or use a calculator to find the probability of a Z-score greater than 0.9428

step 7

Subtract this probability from 1 to find the probability that a diner will have to wait more than 30 minutes

[question 1] Answer

The probability that a diner will have to wait more than 30 minutes for their meal to be served is approximately 0.1729.

Key Concept

Sum of Independent Normal Variables

Explanation

When adding two independent normal variables, the means add and the variances add. The resulting distribution is also normal with the new mean and standard deviation.

Solution by Steps

step 1

Define the total voltage as the sum of two independent normally distributed variables: voltage of type A and type B batteries

step 2

Calculate the mean of the total voltage by adding the means of type A and type B batteries:

5.0 + 8.0 = 13.0

volts

step 3

Calculate the variance of the total voltage by adding the variances of type A and type B batteries:

0.0225 + 0.04 = 0.0625

step 4

Calculate the standard deviation of the total voltage by taking the square root of the variance:

\sqrt{0.0625} = 0.25

volts

step 5

Convert the problem to a standard normal distribution problem by finding the Z-score for 13.4 volts:

Z = \frac{13.4 - 13.0}{0.25} = 1.6

step 6

Look up the Z-score in the standard normal distribution table or use a calculator to find the probability of a Z-score greater than 1.6

step 7

Subtract this probability from 1 to find the probability that the combined voltage exceeds 13.4 volts

[question 2] Answer

The probability that the combined voltage exceeds 13.4 volts is approximately 0.0548.

Key Concept

Sum of Independent Normal Variables

Explanation

The combined voltage of two independent batteries is also normally distributed, with the mean and standard deviation calculated by summing the individual means and variances, respectively.

Solution by Steps

step 1

Define the difference in scores as a normally distributed variable since the scores are independent and normally distributed

step 2

Calculate the mean of the difference by subtracting the English mean from the mathematics mean:

63 - 68 = -5

step 3

Calculate the variance of the difference by adding the variances of the two scores:

10^2 + 7^2 = 149

step 4

Calculate the standard deviation of the difference by taking the square root of the variance:

\sqrt{149} = 12.2066

step 5

Convert the problem to a standard normal distribution problem by finding the Z-score for a difference of 0:

Z = \frac{0 - (-5)}{12.2066} = 0.4095

step 6

Look up the Z-score in the standard normal distribution table or use a calculator to find the probability of a Z-score greater than 0.4095

step 7

This probability is the same as the probability that a student's mathematics score is higher than their English score

[question 3] Answer

The probability that a student's mathematics score is higher than their English score is approximately 0.3413.

Key Concept

Difference of Independent Normal Variables

Explanation

When subtracting one normal variable from another, the means subtract and the variances add. The resulting distribution is also normal with the new mean and standard deviation.

Solution by Steps

step 1

Define the clearance as the difference between the inner diameter of component B and the outer diameter of component A, which are independent normally distributed variables

step 2

Calculate the mean of the clearance by subtracting the mean of component A from the mean of component B:

0.428 - 0.425 = 0.003

step 3

Calculate the variance of the clearance by adding the variances of component A and component B:

0.0001 + 0.0004 = 0.0005

step 4

Calculate the standard deviation of the clearance by taking the square root of the variance:

\sqrt{0.0005} = 0.0224

step 5

Convert the problem to a standard normal distribution problem by finding the Z-score for a clearance of 0:

Z = \frac{0 - 0.003}{0.0224} = -0.1339

step 6

Look up the Z-score in the standard normal distribution table or use a calculator to find the probability of a Z-score less than -0.1339

step 7

This probability is the same as the probability that component A will not fit inside component B

[question 4] Answer

The probability that component A will not fit inside component B is approximately 0.4468.

Key Concept

Difference of Independent Normal Variables

Explanation

The clearance between two components is normally distributed, with the mean and standard deviation calculated by subtracting the means and adding the variances of the individual components' dimensions.