**STAT20029 (T2,2020) Questions: ****Week 7**

**Normal approximation to the binomial distribution (Refer Ch.6)**

**Practice Problem6.13**

PP6.13: Determine whether the following binomial distributions can be adequately approximated by a normal distribution.

(a) p = .1, n = 10

(b) p = .2, n = 30

(c) p = .6, n = 15

**Practice Problem 6.14**

PP6.14: If each of the following binomial distributions can be approximated by a normal distribution, what are the values of the mean and standard deviation?

(a) p = .2, n = 25

(b) p = .5, n = 25

**Review Problem 6.5**

RP6.5: A survey showed that one in five people aged 16 years or older do some volunteer work. If this figure holds for the entire population, and if a random sample of 150 people aged 16 years or older is taken, what is the probability that more than 40 of them do some volunteer work?

**Activity 7.1**

Activity 7.1: The international Data corporation reports that Compaq is number one in PC market share in Hong Kong with 16% of the market. Suppose a researcher randomly selects 130 recent purchasers of PCs

(a) What is the probability that between 15 and 23(both inclusive) PC purchasers bought a Compaq

(b) What is the probability that fewer than 12 PC purchasers bought a Compaq

(c) What is the probability that exactly 22 PC purchasers bought a Compaq

**Activity 7.2**

Activity 7.2: Anthony works in a production plant. Due to the balance of speed and accuracy in production, each part off the line has a 98% probability of defect free production. What is the probability that Antony will produce at least 985 parts without a defect in a 1000 part run? Use normal approximation of the binomial distribution with continuity correction.

**Sampling Distributions (Refer Ch7)**

**Practice Problem 7.9(a)**

PP7.9: A population has a mean of 150 and a standard deviation of 21. If a random sample of 49 is taken, what is the probability that the sample mean is: (a) greater than 154

**Practice Problem 7.10(a) **

PP7.10(a): A population is normally distributed, with a mean of 14 and a standard deviation of 1.2. What is the probability of each of the following? (a) taking a sample of 26 and obtaining a sample mean of 13.7 or more

**Practice Problem 7.13 **

PP7.13: According to the ABS, the average length of stay by tourists in Queensland’s hotels, motels and serviced apartments is 2.2 days, with a standard deviation of 1.2 days. A sample of 65 randomly selected tourists is taken.

(c) What is the probability that the average length of stay for the 65 chosen in the sample is between 2.4 and 3 days?

(d) What would be the minimum value for the average length of stay, for the chosen sample of 65, for the 5% of tourists who stayed longest?

**Practice Problem 7.16 **

PP7.16: A recent survey of Australians found that mean sleep time was 8 hours, with a standard deviation of 3 hours. Assume sleep time is normally distributed.

(a) What is the probability that a randomly selected population member sleeps more than 9 hours?

(b) What is the probability that, in a random sample of 42 people, the sample mean is more than 9 hours?

**Activity 7.3**

Activity7.3: The sign on the lift in a building states “Maximum capacity 1140 kg or 16 persons”. A statistics practitioner wonders what the probability is that 16 people would weigh more than 1140 kg. If the weights of the people who use the lift are normally distributed, with a mean of 70 kg and a standard deviation of 10 kg, what is the probability the statistics practitioner seeks?

**Activity 7.4**

Activity7.4: The persons seeking employment in a management consulting firm have to take a psychometric test given by the firm. From historical data it is found that the scores of the applicants are normally distributed with a mean of 70 and a standard deviation of 10.

(a) If one applicant is selected at random, what is the probability that his or her score will be between 60 and 75?

(b) In a random sample of 16 test scores, what is the probability that the sample mean will be between 66 and 70?

(c) The mean of these 16 test scores has the probability of 0.0668 of exceeding a certain number. What is that number?

**Practice Problem 7.22 **

PP 7.22: According to a survey by Accountemps, 48% of executives believe that employees are most productive on Tuesdays. Suppose 200 executives are randomly surveyed.

(a) What is the probability that fewer than 90 of the executives believe employees are most productive on Tuesdays?

(b) What is the probability that more than 100 of the executives believe employees are most productive on Tuesdays?

**Practice Problem 7.23**

PP 7.23(a): A survey asked business travellers about the purpose of their most recent business trip. Nineteen per cent responded that it was for an internal company visit. Suppose 950 business travellers are randomly selected.

(a) What is the probability that more than 25% of the business travellers say that the reason for their most recent business trip was an internal company visit?

(b) What is the probability that between 15% and 20% of the business travellers say that the reason for their most recent business trip was an internal company visit?

**Activity 7.5**

Activity 7.5: A study at a metropolitan campus of an Australian university reveals that 44% of the students are international students. If a random sample of size 80 is selected, find the probability that more than half of the students in the sample will be international students.

**Practice Problem 7.12(a)**

PP7.12(a): Find the probability in each of the following cases. Consider each population as being finite and therefore apply the finite population correction factor.

(a) P( < 76.5) if N = 1000, n = 60, µ = 75 and ϭ = 6

**Practice Problem 7.14 **

PP7.14: A new estate contains 1500 houses. A sample of 100 houses is selected randomly and evaluated by a real estate agent. If the mean appraised value of a house for all houses in this area is $300 000, with a standard deviation of $10 500, what is the probability that the sample average is greater than $303 000?

**Activity 7.6**

Activity 7.6: The maternity ward of a rural hospital has recorded 520 newborns in a year. A medical researcher selects a random sample of 30 newborns and finds that the mean weight of a newborn is 2.92 kg with a sample standard deviation of 0.85 kg. Estimate the probability that the mean weight of a sample of 30 newborns will exceed 3.00 kg. Use finite population correction factor and assume normality.

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