Could you please provide a step by step solutions (and make sure to write out the equations you will be using to solve it) for this homework.
ST107 Exercise 5 In this exercise you will practise using the normal distribution to approximate the binomial distribution, and consider aspects of sampling distributions of statistics. Question 1 requires the calculation of probabilities under the assumption about the value of a population proportion. Think about whether observed sample data are consistent with the assumption – if not, what would you infer? (This has links with hypothesis testing, covered in Chapter 8.) Questions 2 and 3 consider the sampling distribution of X̄, in particular pay close attention to the effect of the sample size in Question 3. Finally, Question 4 deals with the sampling distribution of a sample proportion – see Section 5.7 of the course pack. 1. A marketing manager believes that 42% of the adult population of London prefer to grind their own coffee beans. A questionnaire on coffee preferences is answered by a random sample of 400 adults. (a) If the 42% assumption is true, find the probability that more than 185 people in the sample prefer to grind their own coffee beans. (b) It turns out that only 145 adults in the sample prefer to grind their own coffee beans. i. Find the probability, if the marketing manager’s assumption was correct, of getting less than or equal to this number (i.e. 145 or fewer) in the sample. ii. On the basis of this probability and the sample result, what do you think of the marketing manager’s claim? [Hint: use a suitable approximation in determining the probabilities.] 2. When a production process is operating correctly, the number of units produced per hour has a normal distribution with a mean of 65 and a standard deviation of 5.2. A random sample of 25 different hours was taken. (a) Find the mean of the sampling distribution of the sample means. (b) Find the variance of the sample mean. (c) Find the standard error of the sample mean. (d) What is the probability that the sample mean exceeds 67 units? 1 3. The fuel consumption, in miles per gallon, of all cars of a particular model has a mean of 49 and a standard deviation of 2. The population distribution can be assumed to be normal. A random sample of these cars is taken. (a) Find the probability that the sample mean of fuel consumption will be less than 47 miles per gallon if: i. a sample of 1 observation is taken ii. a sample of 4 observations is taken iii. a sample of 16 observations is taken. (b) Explain why the three answers in (a) differ in the way they do. 4. An employee survey conducted two years ago by a company found that 65% of its employees were concerned about future healthcare benefits. A random sample of 300 of these employees was asked if they were now concerned about future healthcare benefits. Answer the following assuming that there has been no change in the level of concern about healthcare benefits compared to the survey two years ago. (a) What is the standard error of the sample proportion which are concerned? (b) What is the probability that the sample proportion is less than 0.6? (c) What is the upper limit of the sample proportion such that only 4% of the time the sample proportion would exceed this value? 2
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