**STAT20029 (T2,2020): ****Week 6**

**Normal Distribution**

**Activity 6.1**

Find the area

(a) between ( 0 < z < 1.5 )

(b) between (- 1.75 < z < 0 )

(c) P(z > 1.11 )

(d) P(Z < 1.65)

**Practice problem 6.5(e)**

PP6-5: Determine the probability for the following intervals of the standard normal distribution.

(e) –2.05 < z ≤ –0.87

**Practice problem 6.6**

PP6-6: Determine the probabilities for the following normal distribution problems.

(a) µ = 604, σ = 56.8, x ≤ 635

(c) µ = 111, σ = 33.8, 100 ≤ x < 150

(f) µ = 156, σ = 11.4, x ≥ 170

**Practice problem 6.6**

PP6-6: Determine the probabilities for the following normal distribution problems.

(a) µ = 604, σ = 56.8, x ≤ 635

(c) µ = 111, σ = 33.8, 100 ≤ x < 150

(f) µ = 156, σ = 11.4, x ≥ 170

**Practice problem 6.9(a)**

P6-9: Consider the average home mortgage in New Zealand of $283 000 where the standard deviation of the mortgages is $50 000 and home mortgages are normally distributed. (a) What proportion of home loans are more than $250 000?

**Practice problem 6.11**

P6-11: You are working with a data set where the variable is normally distributed, with a mean of 200 and a standard deviation of 47. In each of the following cases, determine the value of x1.

(a) 60% of the values are greater than x_{1}.

(b) 17% of the values are greater than x_{1}.

**Activity 6.2**

Activity 6.2: The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 264 days and standard deviation 16 days. Doctors agree that serious thought to labor-inducing procedures ought to accompany any pregnancy that runs in the longest 2% of all pregnancies. At what day should these procedures be considered?

**Activity 6.3**

Activity 6-3: An automatic soft drink dispensing machine is supposed to fill 300ml cups with 275ml of soft drink. Due to machine tolerance, the amount actually filled is normally distributed with a standard deviation of 12 ml.

(i) What proportion of cups will fill between 270 ml and 290 ml?

(ii) What proportion of cups will overflow?

**Activity 6.4**

Activity 6-4: The maintenance department of a city’s electric power company finds that it is cost-efficient to replace all street-light bulbs at once, rather than to replace the bulbs individually as they burn out. Assume that the lifetime of a bulb is normally distributed, with a mean of 5300 hours and a standard deviation of 400 hours. If the department wants no more than 2% of the bulbs to burn out before they are replaced, after how many hours should all the bulbs be replaced?

**Activity 6.5**

Activity 6-5: The manager of a surveying company believes that the average number of phone surveys completed per hour by her employees has a normal distribution. She takes a sample of 15 days output from her employees and determines the average number of surveys per hour on these days. The data is:

10.0, 11.2, 11.4, 12.5, 12.2, 12.0, 11.5, 11.7, 11.8, 10.1, 10.3, 10.5, 10.7, 12.2, 15.0

Construct a normal probability plot and infer if the data can be considered as normally distributed.

**Uniform Distribution**

**Practice problem 6.16**

PP6-16(a):. The random variable x is uniformly distributed between 8 and 21.

(a) What is the value of f(x) for this distribution?

(b) Determine the mean and standard deviation of this distribution.

(c) Probability of (10 ≤ x < 17) = ?

**Practice problem 6.15**

PP6-15(a): The random variable x is uniformly distributed between 200 and 240.

(a) What is the value of f(x) for this distribution?

(b) Determine the mean and standard deviation of this distribution.

(c) What is the probability of (x > 230)?

(e) What is the probability of (x ≤ 225)?

**Activity 6.6**

Activity 6-6: The amount of time it takes for a student to complete a statistics quiz is uniformly distributed between 45 and 65 minutes. The lecturer wants to reward (with bonus marks) students who are in the lowest quarter of completion times. What completion time should he use for the cut-off for awarding bonus marks?

**Activity 6.7**

Activity 6-7: A movie starts at 6:00pm, 6:15pm and 6:30pm and that you get to the movie theatre at random between 6:00pm and 6:30pm due to uncertainty in traveling time. What is the probability that you will have to wait for 5 minutes or less before the movie starts? The movie theatre has a policy of not allowing any entry after the movie starts and, assume uniform distribution.

**Activity 6.8**

Activity 6-8: The length of time a student counsellor takes to counsel a university student on his or her personal problems is uniformly distributed over the interval 10 to 45 minutes, inclusively (10 ≤ X ≤ 45). What is the probability that a counselling session will last for more than half-an-hour?

**Exponential Distribution**

**Practice problem 6.21**

PP6-21: Determine the mean and standard deviation of the following exponential distributions.

(a) λ = 3.25?

(b) λ = 0.7?

**Practice problem 6.22**

PP6-22: Let x follow an exponential distribution. Determine the following probabilities.

(a) P (x ≥ 5 | λ = 1.35)

(d) P (x < 6 | λ = 0.8)

**Practice problem 6.23**

PP6-23: The average length of time between arrivals at a motorway tollbooth is 23 seconds. Assume that the time between arrivals at the tollbooth is exponentially distributed.

(a) What is the probability that a minute or more will elapse between arrivals?

**Practice problem 6.24**

PP6-24: A busy restaurant determined that between 6.30 pm and 9.00 pm on Friday nights, the arrival of customers is Poisson distributed with an average arrival rate of 2.44 per minute.

(a) What is the probability that at least 10 minutes will elapse between arrivals?

(d) What is the expected length of time between arrivals?

**Activity 6.9**

Activity 6.9: On Saturdays, cars arrive at Hippos Car Wash at the rate of 4 cars per fifteen minute intervals. Using the exponential distribution, determine

(a) the probability that at least 5 minutes will elapse between car arrivals.

(b) What is the mean of this exponential distribution?

**Activity 6.10**

Activity 6.10: Customers access a vending machine at the rate of 12 per hour. Using the exponential distribution, determine

(a) the probability that at least 10 minutes will elapse between accesses,

(b) the standard deviation of this exponential distribution.

**Activity 6.11**

Activity 6.11: A catalogue company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 2.8 minutes. What is the probability that a randomly selected caller is placed on hold fewer than 5 minutes?

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