STAT217 Midterm practice questions
Note: In addition to these questions, you should be able to answer the questions on worksheets/tutorial sheets 1 through 3.
Question 1
a)
For an experimental vehicle, past records indicate the
distance the vehicle can travel is normally distributed with a mean of 1000 km
and standard deviation of 250 km. A modification to the engine was made to see
if the average distance travelled could be significantly increased. After the
modification, 14 trials were conducted; the average distance traveled was 1120
km. Test at a 5% level of significance.
b) For
which levels of significance between 1% and 10% would the opposite conclusion
be reached? (between 1% and 3.59%)
c)
If the actual average distance traveled with the modified
engine is 1120 km, what is the power of the test at a 5% level of significance?
(55.96%)
Question 2
a) An environmental watchdog wanted to see if the percentage of companies exceeding water pollution limits is significantly less than 2%. A survey of 500 companies indicated that 6 of them exceeded the limits. Test at a 1% level of significance.
b) What
would be the power of the test if the actual percentage of companies exceeding
water pollution limits is 1.2% at a 5% level of significance? (31.92%)
Question 3
A tourist magazine wanted to compare inner city hotels and those in the suburbs to see if there was any significant difference in average daily rates. It sampled 12 from each type. These were the results:
|
Downtown |
597 |
400 |
602 |
309 |
292 |
458 |
452 |
484 |
492 |
753 |
421 |
777 |
|
Suburbs |
429 |
531 |
475 |
519 |
479 |
458 |
469 |
499 |
468 |
482 |
435 |
472 |
Analysis of the data indicates both groups are normally distributed but do not share equal variances.
a) Test at a 5% level of significance.
b) The p-value for the above test is 56.66%. If a level of significance had not been chosen, why would the same conclusion be reached? (p-value > 10%)
c) Construct a 95% confidence interval of the average difference in average daily rates between downtown and suburban hotels. Round the margin of error to the nearest cent. If this interval were used to test the hypothesis in part a, why would the same conclusion be reached? (-72.91 < m1 - m2 < 126.41)
Question 4
A focus group of 10 IT professionals were asked to compare two operating systems on a scale from 1 to 10 with 10 being best. These were the results:
|
Person |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
System A |
5 |
7 |
7 |
4 |
6 |
7 |
7 |
7 |
7 |
6 |
|
System B |
8 |
7 |
6 |
7 |
6 |
8 |
9 |
8 |
8 |
8 |
Is system B rated significantly better than system A? Test at a 5% level of significance.
Question 5
An outreach worker estimates that the average weekly income of a panhandler is significantly less than $45 but has no prior information to work with. A survey has the following results:
|
40.95 |
46.56 |
41.58 |
37.96 |
47.35 |
31.19 |
35.35 |
Analysis of the data indicates that it is normally distributed.
a) Test the hypothesis at a 5% level of significance.
b) The p-value for the above test is 3.46%. For which levels of significance between 1% and 10% would we conclude that the average income is not significantly less than $45?
Question 6
An educational researcher wanted to determine if the average grade 12 achievement score was significantly higher for one region than for a second region which has a higher percentage of students whose second language is English. These were the results:
|
|
Region 1 |
Region 2 |
|
Mean |
67.5 |
64.4 |
|
Standard deviation |
7.8 |
8.2 |
|
Sample size |
100 |
100 |
a) Test the hypothesis without specifying a level of significance.
b) Construct a 95% confidence interval of the average difference in the scores between the 2 regions. Round the margin of error to 1 decimal. (0.9 < m1 - m2 < 5.3)
Question 7
A company produces custom wooden crates that are sold to shipping companies worldwide. Each crate is assembled by skilled line-operators. Each operator is expected to produce, on the average, the same number of finished units per day. Data for two operators was collected for a period of a week (6 days total) and resulted in the following data:
|
Day |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
Saturday |
|
Operator 1 |
108 |
115 |
118 |
116 |
110 |
115 |
|
Operator 2 |
106 |
118 |
116 |
115 |
114 |
112 |
Analysis of the data indicates it is normally distributed.
a) Is there any significant difference in average production between the two operators? Test at a 5% level of significance.
b) Construct a 95% confidence interval of the average difference between the two operators. Round the limits to 1 decimal. If this interval were used to test the hypothesis in part a, why would the same conclusion be reached? (-2.9 < md < 3.2)
Question 8
A marketing research analyst has been hired by a major consumer electronics manufacturer to study the difference in age usage of MP3 systems. The two groups studied were those 25 years old and younger and those over 25 years of age. In the group of individuals 25 years and younger a sample of 1200 individuals were surveyed and 900 expressed that they used MP3s on a regular basis. The group of individuals that were over 25 years of age was made up of 1500 individuals and 900 of them expressed that they used MP3s on a regular basis.
a) Is the difference in the percentage of the two age groups that use MP3s on a regular basis more than 10%? Test at a 5% level of significance.
b) Suppose a level of significance had not been chosen. Why would the same conclusion be reached? (p-value < 1%)
Question 9
A restaurant was usually slow on Tuesday and Wednesday
nights. It introduced specials on these night to attract more customers. Sales
are normally distributed with average daily sales of $9200 and standard
deviation of $1300. A sample of 12 days of data had the following results:
|
7514 |
9508 |
11966 |
10389 |
|
10206 |
10040 |
10767 |
9251 |
|
8805 |
10947 |
8928 |
9834 |
a)
If the goal is to have significantly less variability
in daily sales, test the hypothesis at a 1% level of significance.
b) Construct a 99% confidence interval of the standard deviation of the daily sales. Round the limits to the nearest cent.
Question 10
A study is being conducted by a major marketing research firm with regard to the number of full-page advertisements run in nationally circulated monthly magazines. They are particularly interested in two of these magazines, A and B. A random sample of monthly issues was selected and the number of full-page ads was counted and resulted in the following data.
|
A |
17 |
26 |
18 |
16 |
15 |
23 |
20 |
|
B |
8 |
9 |
24 |
12 |
10 |
|
|
Analysis of the data indicates that both groups are normally distributed with equal variances.
a) Does magazine A have significantly more full-page ads on average than magazine B? Test at a 5% level of significance.
b) The p-value for the above test is 2.57%. For which levels of significance between 1% and 10% would we conclude that magazine A has significantly more full-page ads than magazine B? (2.57% to 10%)
c) Construct a 95% confidence interval of the average difference between magazine A and magazine B in the average number of full-page ads. Round the limits to 1 decimal. (-0.1 < m1 - m2 < 13.4)
Question 11
In a focus group, people from two economically different neighbourhoods were asked how often they eat out in a month. They sampled 6 people from each type of neighbourhood. The results were:
|
Person |
1 |
2 |
3 |
4 |
5 |
6 |
|
Richer |
17 |
18 |
18 |
19 |
18 |
5 |
|
Poorer |
5 |
9 |
9 |
8 |
4 |
5 |
Analysis of the data indicates that the richer neighbourhood is not normally distributed. Do the people in the richer neighbourhood eat out more often on average than those in the poorer neighbourhood? Test at a 5% level of significance.