MGMT2262 Tutorial Sheet #5

Sales at a particular store are normally distributed with a mean of $30 and a standard deviation of $12.50.

1) If a sample of 50 sales is taken, what is the probability the average of this sample is less than $25? (0.0023)

2) Suppose a sample of 50 individual sales is taken and the mean of this sample is $29.50. Construct a 95% confidence interval of the average individual sale at this store. ($26.04 < m < $32.96)

3) Suppose they increase the level of confidence to 99% from 95%. How much does the margin of error increase by? ($1.09)

4) Suppose they want to conduct another study. What should be the sample size if they use the accepted standard deviation of $12.50 and they want the margin of error to be no more than $2.50 and a level of confidence of 95%? (97)

5) The percentage of CO₂ in Fizzy Soda follows a Beta distribution with a mean of 75% and a standard deviation of 1.2%. Each hour, a sample of 60 cans is taken and the amount of CO₂ is measured for quality control purpose. They want the percentage of CO₂ to be between 74.7% and 75.3%. In a sample of 60 cans, what is the probability of that happening? (0.9476)

A social service agency devoted to helping mentally handicapped children learn wanted to explore the efficiency of a new teaching method for helping children learn basic addition and subtraction. Since it was a new method, the agency decided to start from scratch. The method was tried on a group of 8 students. At the end of the program, the students’ average success rates were recorded:

69.6

61.8

74

82.2

81.6

85.9

54.5

70.1

Analysis of the data indicated it is normally distributed.

6) Construct a 95% confidence interval of the average success rate based on this data, rounding to 1 decimal. (63.5 < m < 81.5)

7) If the level of confidence is increased to 99%, what is the confidence interval now? (59.14 < m < 85.78)

8) A convenience store wanted to see if people spend more than $5 on average. They took a sample of 60 customers and the average was $5.30 with a standard deviation of 1.25. Do people spend significantly more than $5 on average? Test at a 5% level of significance. (Z = 1.86; reject Ho; conclude people spend more than $5 on average)

9) Is there a level of significance between 1% and 10% in which you would have reached the opposite verdict? (p-value = 0.0314; any level of significance between 1% and 3.14%)

10) A researcher wanted to see if there was any change in the average amount of time that people spend commuting to work compared to 5 years ago. Based on data compiled 5 years ago, the amount of time people spend commuting to work is normally distributed with an average of 25 minutes with a standard deviation of 14.2 minutes. A survey of 200 people showed these people spend 27.9 minutes on average traveling to work. Test the hypothesis at a 1% level of significance. (Z = 2.89; conclude that people no longer spend 25 minutes on average commuting to work)

11) Suppose a level of significance had not been chosen in question 10. Why would the same conclusion be reached? (p-value = 0.0038; p-value is less than 1% under general rule of thumb)

12) A vendor was concerned the coffee machine was dispensing less than 175 ml into a cup. Past records indicate the amount of coffee in a cup is normally distributed with a standard deviation of 2.05 ml. A sample of 12 cups was taken and the average of this sample was 173.5 ml. Is the average amount of coffee in a cup significantly below 175 ml? Use the p-value approach to reach your decision. (Z = -2.53; p-value = 0.0057; reject Ho; conclude average amount is less than 175 ml)

13) A new clock system was being tested against Greenwich Mean Time (GMT) to see if there was any significant difference between the two. Six different times were compared with the following differences in seconds:

0.002	0.05	-0.008
-0.0003	0.0032	-0.015

If they assume the above differences follow a normal distribution, is there any significant difference between the time on the clock and GMT? Use the p-value approach. (t=0.5674; p-value > 0.1; do not reject Ho; conclude no significant difference between times)

14) Construct a 95% confidence of the average difference between the clock and GMT, rounding to 2 decimals and interpret it. If the hypothesis in question 13 had been tested at a 5% level of significance, why would you reach the same conclusion? (-0.02 < m < 0.03; with 95% confidence, the average difference between the clock and GMT is between –0.02 and +0.03 seconds; we would reach the same conclusion because the hypothesis mean of zero is in the confidence interval)