Statistics and Probability Answers

Below are the number of individuals in each department

Department

Numbers

Accounts

18

Marketing

24

Sales

43

Repairs

35

How many employees from the Marketing department should Caitlyn select to be part of the sample? [2]

v. Another employee suggested that Caitlyn should group employees by the type of car they own. State a reason why grouping the employees in this way will NOT be appropriate to select this sample using the sampling method in ii

A psychiatrist is testing a new antianxiety drug, which seems to have the potentially harmful side effect of lowering the heart rate. For a sample of 50 medical students whose pulse was measured after 6 weeks of taking the drug, the mean heart rate was 70 beats per minute (bpm). If the mean heart rate for the population is 72 bpm with a standard deviation of 12, can the psychiatrist conclude that the new drug lowers heart rate significantly  (Set the level of significance to 0.01.) 

SOLUTIONS:

Step 1: State the hypotheses.

Ho: 

Ha:

Step 2: The level of significance and the critical region. 𝛼 = _____, 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 = _____. 

Step 3: Compute for the value of one sample test.

𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑡𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 = _______. 

Step 4: Decision rule.

Step 5. Conclusion.

The Gauteng traffic department records show that 25% of all drivers wear seatbelts. In a random sample of 400 cars stopped at a roadblock in Gauteng, 152 of the drivers were wearing seatbelts. A 90% confidence interval for the proportion in the population who wear seatbelts is given by (round final answer to two decimal places):

Employees in a large manufacturing plant worked an average of 62 hours of overtime last year, with a standard deviation of 15 hours. The union president and the human resources director jointly selected a sample of 36 employees and had discussions on the company’s work rules and overtime policies. What is the probability that employees in this sample worked an average of less than 65 hours of overtime?

A psychiatrist is testing a new antianxiety drug, which seems to have the potentially harmful side effect of lowering the heart rate. For a sample of 50 medical students whose pulse was measured after 6 weeks of taking the drug, the mean heart rate was 70 beats per minute (bpm). If the mean heart rate for the population is 72 bpm with a standard deviation of 12, can the psychiatrist conclude that the new drug lowers heart rate significantly? (Set the level of significance to 0.01.)

Solution: Step 1: State the hypotheses.

Ho: _______________________________________________________________

Ha: _______________________________________________________________

Step 2: The level of significance and the critical region. 𝛼 = _____, 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 = _____.

Step 3: Compute for the value of one sample test. 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑡𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 = _______.

Step 4: Decision rule. ____________________________________________________________

Step 5. Conclusion. ______________________________________________________________

A milk processing company test implemented a plant-wide energy conservation program with a goal of reducing the mean daily consumption rate of at least 1,000 kWh from its normal operating plants. The conservation program was implemented in Plant B. The following data was collected on weekdays where consumption level is at its peak. Were the conservation efforts effective in achieving its goal? Compare the results with Plant A’s data where the program is not implemented using 𝛼 = 0.05. Assume that the population variances are not equal.

 Plant A (kWh) 3,952.80 3,276.00 3,636.00 3,636.00 3,636.00 3,636.00 4,068.00 4,068.00 4,362.00 4,362.00 4,362.00 4,362.00 3,882.00 3,808.80 3,808.80

Plant B (kWh) 4,036.00 4,036.00 4,036.00 3,264.00 864.00 1,368.00 2,196.00 4,392.00 5,220.00 3,600.00 3,960.00 4,428.00 756.00 612.00 684.00

The duration of the AstraZeneca Covid-19 vaccine side effect is normally distributed with a mean of 1.5 days and a standard deviation of 0.4 days.

(i) If a AstraZeneca Covid-19 vaccine recipient is selected at random, find the percentage

that the duration of side effect is less than 24 hours. (3 marks)

(ii) 96% of the AstraZeneca Covid-19 vaccine recipients have experienced more than k days of discomfort symptoms. Find the value of k. (3 marks)

(iii) 7 AstraZeneca Covid-19 vaccine recipients are selected randomly, find the probability

that 3 recipients have less than 24 hours of side effect.

The number of AstraZeneca Covid-19 vaccine recipients who had severe discomfort symptoms

found to be 15 per 1000 population. Assuming the number of people who received the

AstraZeneca Covid-19 vaccine had severe discomfort symptoms follows a Poisson distribution.

ii) Find the probability that in a randomly selected 200 population, there are 4 or 5 people

who received the AstraZeneca Covid-19 vaccine have severe discomfort symptoms.

i) Find the probability that in a randomly selected 15000 population, there are at least 181

people who received the AstraZeneca Covid-19 vaccine have severe discomfort

symptoms.