Statistics and Probability Answers

Most production processes generate defectives. However, the percentage of defectives will usually be kept as low as possible. To check the production process, random samples or products areusually drawn and tested.

Consider a certain mass-produced item. Suppose that the production process yields at most 2% defectives, at least when the process is what is termed ‘under control’. To check the quality of the production process, 400 of the items are randomly selected each hour. If at most 9 defectives are detected, then it is concluded that the process is, still under control and nothing is done. But if 10 or more defectives are found, then it is concluded that the production process is, in the terminology, ‘out of control’, and the process is stopped and investigated. Below, p denotes the proportion of defectives in the large hourly production and bp denotes the (random) sample proportion of the defectives in an hourly sample of size 400.

(a) Determine the probability distribution of p

A consumer organisation and the manufacturer of a certain brand of margarine are in dispute. The manufacturer states that his packs of margarine contain at least 250g of margarine whereas the consumer organisation claims that the content is at most 249g. A judge has to decide the issue.The judge decides to base her judgement on a random sample of ten packs of margarine.

The consumer organisation will be adjudged to be right if the random sample has a mean weight of less than 249:5g.

The manufacturer will be adjudged to be right if this mean weight is at least 249:5g.

The starting point for the judge is that he assumes a normal distribution with mean and variance 4, that is (u; 4) to be a good model for the weight X of a pack of margarine of that brand. Below, X denotes the mean weight (in grams) of the ten packs of margarine.

a)Determine the probability distribution of X

b)Calculate the probability that the judge decides (incorrectly) that the consumer organisation is right while the true u is equal to 251.

For the production process of ballpoint pens of a certain type, it is known that 5% of the pens are defective. The manager believes that this percentage is too high and hence changes the production process. To test the merits of this new process, a random sample of 350 pens is taken; only 9 of them are defective.

a) (i) Determine a 90% confidence interval for p, the population proportion of defective pens produced with the new process.

ii) Interpret the confidence interval.

iii) Do you think that the new process is better?  

b) Conduct a two-sided test to conclude whether revising the process has indeed changed the population proportion of defective pens. Take a= 0:05.

i) State the null and alternative hypothesis.

ii) State and calculate the appropriate test statistic.

iii) Determine the critical value of the test and state the rejection region.

iv) State whether or not you reject the null hypothesis.

v) Draw an appropriate conclusion.

Ten percent of Americans are allergic to ragweed. Suppose a random sample of 200 people is selected.

4. Evaluate np and n(1 – p). Is it appropriate to use the normal distribution to approximate the binomial distribution? (Round to 2 decimal places)

            np =                             n(1 – p) =                         Enter Y or N

Ten percent of Americans are allergic to ragweed. Suppose a random sample of 200 people is selected.

1. Identify n    

2. Identify p   (Round to 2 decimal places)

3. Identify 1 – p   (Round to 2 decimal places)

Given that𝜇=15  and 𝜎=3

Suppose samples of size five are taken. 

1. What is the mean of the sampling distribution?    

2. What is the standard deviation of the sampling distribution?    

(Round answers to TWO decimal places)

Suppose that the sampling distribution of size 10 has a mean of 30 and standard deviation of 3. Calculate

μ =  

σ=  

Suppose a survey by the Department of Roads in 2019 revealed that 60% of the vehicles travelling on E to F Highway, where speed limits are posted at 80 kilometres per hour are found to be exceeding the limit. Suppose you randomly record the speed of 10 vehicles travelling on the highway as part of your internship. The sample size of the entire survey is not disclosed to you.

Compute the following probabilities. Showing all working.

a) P (X = 2).

b) P (X = 5).

c) P (X = 10).

A random sample of high school students are asked the number of hours they spent on social media. The hours are reported below: 3 5 6 7 8 10 12 13 14 15 You can assume the times are normally distributed with a standard deviation of 8.5.

a) Estimate with 90 percent confidence the mean number of hours spent on social media by all high school students. Show all working as part of your answer and interpret your answer.

Z=0.97, find the are between z=0

A questionnaire is being devised to collect data from a sample of single South African adults to answer the following two research questions:

A. Are single men and single women equally likely to prefer cats and dogs as pets?

B. Is there a difference between single South African men and women in how many pets they own?

Assess each of the questions below for inclusion in the questionnaire: Should the question be included or not? Explain your choice by discussing the necessity, relevance, clarity and lack of bias of each of the suggested questions. A question should only be included if it satisfies all these criteria, and you should only include questions which are needed to answer the two research questions. (Note that you need to assess the questions as a researcher. We are not asking you to answer these questions yourself!)

a) please indicate your gender (options: male/ female)