Statistics and Probability Answers

Develop R-code for the following questions.
1. According to a dietary study, a high sodium intake may be related to ulcers,
stomach cancer, and migraine headaches. The human requirement for salt is only 220 milligrams per day, which is surpassed in most single servings of ready-to-eat cereals. If a random sample of 20 similar servings of certain cereal has a mean sodium content of 244 milligrams and a standard deviation of 24.5 milligrams, does this suggest at the 0.05 level of significance that the average sodium content for a single serving of such cereal is greater than 220 milligrams? Assume the distribution of sodium contents to be normal.

3. A certain geneticist is interested in the proportion of males and females in the population that have a certain minor blood disorder. In a random sample of 1000 males, 250 are found to be afflicted, whereas 275 of 1000 females tested appear to have the same disorder. Test the difference between the proportion of males and females that have the blood disorder at 1% level of significance. Develop R- code to justify.

2. The mean life time of a sample of 400 fluorescent light bulbs produced by a company is found to be 1,570 hours with a standard deviation of 150 hours. Develop an R-code to test the hypothesis that the mean life time of bulbs is 1600 hours against the alternative hypothesis that it is greater than 1,600 hours at 1% and 5% levels of significance. Draw your conclusions with the results.

1. Your statistics instructor claims that 60 percent of the students who take his Elementary Statistics class go through life feeling more enriched. For some reason that he can't quite figure out, most people don't believe him. You decide to check this out on your own. You randomly survey 64 of his past Elementary Statistics students and find that 34 feel more
enriched as a result of his class. Develop an R-code to draw your conclusions.

Solve using R:
1. The weekly wages of 1000 workmen are normally distributed around a Mean of Rs. 70 with a standard deviation of Rs.5. Develop an R-code to estimate the number of workers and
visualize the area under normal curve whose weekly wages will be (i) between Rs.60 and Rs.80, (ii) less than Rs 65, (iii) More than Rs. 75.


2. It is conjectured that an impurity exits in 10% of all drinking wells in a certain rural community. In order to gain some insight on this problem, it is determined that some tests should be made. It is too expensive to test all of the many wells in the area, so 20 were selected at random for testing. Write R-code to find and visualize the probability that (i) exactly 2 wells have the impurity? (ii) more than 4 wells are impure? (iii) between 3 and 6 wells, inclusive, are impure?

The price of a popular tennis racket at a national chain store is $179. Portia bought ten of the same racket at an online auction site for the following prices: 155, 179, 175, 175, 161, 158, 170, 165, 163 and 172. Assuming that the auction prices of rackets are normally distributed, determine whether there is sufficient evidence in the sample, at the 5% level of significance, to conclude that the average price of the racket is less than $179 if purchased at an online auction.

The survival time, in weeks, of a component X follows an exponential distribution with
parameter β = 5. (i)What is the probability that the survival time will exceed 10 weeks? (ii) What is the conditional probability that the component will survive at least 8 weeks given that it is working at the end of 3rd week? (iii) What is probability that the component will survive between 6 and 12 weeks?

A poll was conducted between town voters and country voters for favoring the certain proposal. If 60 of 100 town voters favor to proposal and 150 of 300 country residents favor it, would you agree that the proportion of town voters favoring the proposal is higher than the proportion of country voters at 5% level of significance?.

In a data distribution, the first quartile, the median and the means are 30.8, 48.5 and 42.0 respectively. If the coefficient skewness is −0.38 , use the information to answer questions 1 and 2
1. What is the approximate value of the third quartile, correct to 2 decimal places?
2. What is the approximate value of the variance, correct to the nearest whole number?

The life time of a certain brand of bulbs produced by a company is normally
distributed, with mean 210 hours and standard deviation 56 hours. What is the
probability that a bulb picked at random from this company’s products will
have a life time of:
(i) at least 300 hours,
(ii) at most 100 hours,
(iii) between 150 and 250 hours.