Statistics and Probability Answers

Suppose we would like to gauge voters’ preferences for the President of Mars’ election. In an exit poll with a random sample of 120 voters, 30 voted for Mark Zuckerberg and 90 voted for Elon Musk. For the following, report your answers in 3 decimal places.

However in some applications, we would like to know whether Elon Musk is getting at least a certain percentage of the total votes. In this case, we should use a 100(1 − α)% lower one-sided confidence interval taking the form of [L, 1]. To construct a 100(1 − α)% lower one-sided Wald confidence interval through the pivot method, we start by setting P   X − p qp(1−p) n < zα   = 1 − α,

where X is the sample proportion, n the sample size and zα is the upper 100α% quantile of N(0, 1). Using this relationship, construct a 98% lower one-sided Wald confidence interval for p.

Suppose we would like to gauge voters’ preferences for the President of Mars’ election. In an exit poll with a random sample of 120 voters, 30 voted for Mark Zuckerberg and 90 voted for Elon Musk. For the following, report your answers in 3 decimal places.

(a) Construct a 98% Wilson score confidence interval for Mars’ population proportion p supporting Elon Musk.

(b) Construct a 98% Wald confidence interval for p.

(c) Explain why Wald confidence intervals can have lower coverage in practice when compared to Wilson score confidence intervals

ILAW Manufacturing company produces bulbs that last a mean of 900 hours with a standard deviation of 110 hours. what is the probability that the mean lifetime of a random sample of 15 of these bulbs is less than 850 hours?

Inside a box are a R10 note, a R20 note, a R50 note and a R100 note. Consider the experiment in which two notes are drawn from the box without replacement. Let 𝑋 be the random variable that gives the value of the smaller (Rands) note. Construct the probability mass function for 𝑋

In an intelligence test conducted on 1000 students, the average scores was 42 and standard deviation 24. 

Find the number of successful students if the minimum pass marks is 40.

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14. Weights of a certain model of fully loaded gravel trucks follow a normal distribution with mean 

 = 6.4 tons and standard deviation  = 0.4 ton.

a) For a truck selected at random, what is the 

probability the weight will be 

less than 5.9 tons?

b) For a truck selected at random, what is the 

probability the weight will be 

more than 7 tons?

c) For a truck selected at random, what is the 

probability the weight will be 

between 5.4 and 6.4 tons?

Take a sample of 60 grades and test the following: 1) Test for the hypothesis that the average grade is not 75.  2) Test for the hypothesis that the average grade below is higher than 75. 3) Calculate the confidence limits for the population mean at 95 %

8.A researcher wants to investigate whether different forms of exercise can be used to help hyperactive children. A group of 90 children are randomly assigned to one of three groups. The first group will just do their normal exercise. The second group will be given an additional exercise routine (moderate). The third group will be given an additional exercise routine (strenuous). At the end of a four-month period parents will be asked to evaluate their children's ability to participate in calm activities.

a)What are the experimental units or subjects?

b)What is the factor (explanatory variable)?

c)What is the response variable?

d)Is this a blind study? Explain.

6. When a random sample of 600 Americans were surveyed, 37% said that their favorite beverage is coffee.

a) Find the margin of error as a percent to the nearest tenth. 

b) If another sample is taken and found that shows 40% reported coffee as their favorite beverage, does this fall in the acceptable range from the first survey?