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Statistics Task on Probability

Statistics Task on Probability

  • 9th Jun, 2022
  • 16:13 PM

1, In an experiment, out of 50 students, 43 submitted an optional document.
Compute a 92% confidence interval for the true proportion of students who are expected to submit their document if the testing phase is believed to represent the actual situation. Interpret the confidence interval.
Use the normal approximation to the binomial to find the probability that in a sample of 500 students more than 40 will submit their document. State whether the use of the approximation is justified by reasoning for your conclusion.
Ans : The probability of more than 40 students will submit the documents out of 500 is 1.
   The probability of more than 40 students will submit the documents out of 50 is 0.0000028070.

2, Additionally, there was a discrepancy between the proportion of males and females who submitted the document. In fact, 18 out of 20 males and 25 out of 30 females submitted their document.
Construct a 93% confidence interval for the difference between the proportion of males and females who are expected to submit their document. Interpret your result.
Ans:    
   H0 : Proportions are same
      Ha : Proportions are not same
      Z = 6.67
      Critical value for 95%  C.I =1.96
      Critical value for 90%  C.I =1.645
       93% C.I value lies between 1.96 and 1.645
       Here statistic value (6.67) is greater than the both values (1.96 and 1.645).
            Therefore we reject H0 (Null Hypothesis).
             Difference exists between proportion of males and proportion of females.
        

3, Male students claimed that it took them 5.2 minutes to submit their document while for females it took 4.7 minutes. Standard deviations stood at 1.2 minutes for males and 1.6 minutes for females. When students were made aware of this, females were teasing their male counterparts that they are much more efficient. 
Construct a 93% confidence interval for the difference between the average time taken by males and females to submit their document and state, by giving reasons to support your conclusion, whether the females’ claim is justified.
Ans: 
       H0 : Girls are efficient than boys
      Ha : Girls are not efficient than boys
      t= 25.21
      Critical value for 95%  C.I = 3.505
      Critical value for 90%  C.I = 3.269
       93% C.I value lies between 3.269 and 3.505
       Here statistic value (25.21) is greater than the both values (3.269 and 3.505).
            Therefore we reject H0 (Null Hypothesis).
             

4, Let Y represent the amount of time (in minutes) students took to upload the document. This random variable is believed to have an exponential distribution where E(Y)=4.
 Find the probability that a student spends between 4 and 5 minutes uploading the document.
 Find the probability that a student spends more than 6 minutes uploading the document.
Ans: 
i)    0.6321
ii)    0.2231
 

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