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saw5 [17]
3 years ago
6

The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes

and a standard deviation of 10 minutes. Answer the following questions:
A) What is the probability of completing the exam in ONE hour or less?

B) what is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes?

C) Assume that the class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to compete the exam in the allotted time?

Mathematics
2 answers:
kotegsom [21]3 years ago
4 0

Answer:

Part A:

Probability is P(z<-2)=1-0.9772=0.0228

Part B:

P(-2<z<-0.5)=0.2857

Part C:

Number of students unable to complete the exam=60-50=10 students

Step-by-step explanation:

Part A:

Mean=μ=80 min

Standard Deviation=σ=10 min

Formula:

z=\frac{\bar x- \mu}{\sigma}

where:

\bar x=60\ min

z=\frac{60-80}{10}\\ z=-2

Probability is P(z<-2)

From the probability distribution tables (Cumulative Standardized normal distribution table)

Probability is P(z<-2)=1-0.9772=0.0228

Part B:

For 75 min:

z=\frac{75-80}{10}\\ z=-0.5

For \bar x=60\ min

z=\frac{60-80}{10}\\ z=-2

From the probability distribution tables (Cumulative Standardized normal distribution table)

P(-2<z<-0.5)=P(z<-0.5)-P(z<-2)

P(-2<z<-0.5)=(1-0.6915)-(1-0.9772)

P(-2<z<-0.5)=0.2857

Part C:

\bar x=90\ min

z=\frac{90-80}{10}\\ z=1

From the probability distribution tables (Cumulative Standardized normal distribution table)

Probability is P(z<1)=0.8413

Number of students=0.8413*60

Number of students to complete the exam=50.478≅50

Number of students unable to complete the exam=60-50=10 students

vesna_86 [32]3 years ago
3 0

Answer:

A) P=2.28%

B) P=28.57%

C) I expect 10 students to be unable to complete the exam in the alloted time.

Step-by-step explanation:

In order to solve this problem, we will need to find the respective z-scores. The z-scores are found by using the following formula:

z=\frac{x-\mu}{sigma}

Where:

z= z-score

x= the value to normalize

\mu = mean

\sigma= standard deviation

The z-score will help us find the area below the normal distribution curve, so in order to solve this problem we need to shade the area we need to find. (See attached picture)

A) First, we find the z-score for 60 minutes, so we get:

z=\frac{60-80}{10}=-2

So now we look for the z-score on our normal distribution table. Be careful with the table you are using since some tables will find areas other than the area between the mean and the desired data. The table I used finds the area between the mean and the value to normalize.

so:

A=0.4772 for a z-score of -2

since we want to find the number of students that take less than 60 minutes, we subtract that decimal number from 0.5, so we get:

0.5-0.4772=0.0228

therefore the probability that a student finishes the exam in less than 60 minutes is:

P=2.28%

B) For this part of the problem, we find the z-score again, but this time for a time of 75 minutes:

z=\frac{75-80}{10}=-0.5

and again we look for this z-score on the table so we get:

A=0.1915 for a z-score of -0.5

Now that we got this area we subtract it from the area we found for the 60 minutes, so we get:

0.4772-0.1915=0.2857

so there is a probability of P=28.57% of chances that the students will finish the test between 60 and 75 minutes.

C) Finally we find the z-score for a time of 90 minutes, so we get:

z=\frac{90-80}{10}=1

We look for this z-score on our table and we get that:

A=0.3413

since we need to find how many students will take longer than 90 minutes to finish the test, we subtract that number we just got from 0.5 so we get:

0.5-0.3413=0.1586

this means there is a 15.86% of probabilities a student will take longer than 90 minutes. Now, since we need to find how many of the 60 students will take longer than the 90 available minutes, then we need to multiply the total amount of students by the percentage we previously found, so we get:

60*0.1586=9.516

so approximately 10 Students will be unavailable to complete the exam in the allotted time.

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Answer:

The 90% confidence interval  -49.8

The null hypothesis is  H_o : \mu_1 = \mu_2

The alternative hypothesis H_a  :  \mu_1 <  \mu_2

The distribution test statistics is t = -3.222

The rejection region is  p-value < \alpha

The decision rule is reject the null hypothesis

The conclusion is

      There is sufficient evidence to conclude that there are more passengers riding the 8:30  train

The p-value  is  p-value  =0.000951

Step-by-step explanation:

From the question we are told that

    The first sample size n_1 = 30

    The first sample mean is  \= x _1 = 323

    The first standard deviation is s_1 = 41

    The second sample size is n_2 = 45

    The second sample mean is  \=x_2 = 356

    The second standard deviation is s_2 = 45

given that the confidence level is 90% then the level of significance is mathematically represented as

          \alpha  = (100 -90)\%

         \alpha  = 0.10

Generally the critical value of \frac{\alpha }{2} obtained from the normal distribution table is  

   Z_{\frac{\alpha }{2} } = 1.645

Generally the pooled variance is mathematically represented as

        s^2 = \frac{(n_1 - 1)s_1^2  + (n_2 -1)s_2^2 }{n_1 + n_2 -2}

      s^2 = \frac{(30 -1)(41^2) + (45-1)45^2}{30+45 -2}

     s^2 = 1888.34

Generally the standard error is mathematically represented as

     SE =  \sqrt{\frac{s^2}{n_1} + \frac{s^2}{n_2}  }

=>  SE =  \sqrt{\frac{1888.34}{30} + \frac{1888.34}{45}  }

=>   SE =  10.24

Generally the margin of error is mathematically evaluated as  

      E =  Z_{\frac{\alpha }{2} } * SE

       E =  1.645* 10.24

       E = 16.85

Generally the 90% confidence interval is mathematically represented as

     \=x_1 -\=x_2 -E < \mu_1 -\mu_2 < \=x_1 -\=x_2 +E

     323 -356 -16.84

     -49.8

The null hypothesis is  H_o : \mu_1 = \mu_2

The alternative hypothesis H_a  :  \mu_1 <  \mu_2

Generally the test statistics is mathematically represented as

     t =  \frac{\= x_1 - \=x_2 }{SE}

=>   t = \frac{323-356}{10.24}

=>   t = -3.222

Generally the degree of freedom is mathematically represented as

     df =  n_1+n_2 -2

      df = 30 + 45 -2

     df = 73

The p-value is obtained from the student t distribution table at degree of freedom of 73 at 0.05 level of significance

    The value is  p-value  =0.000951

Here the level of significance is  \alpha =  5\%  =  0.05

Given that the p-value < \alpha then we  reject the null hypothesis

Then the conclusion is  

  There is sufficient evidence to conclude that there are more passengers riding the 8:30  train

               

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