I need help with these homework

Problem 1 [10 points = 5 + 5]

A sample of n = 16 waiters was selected at random to draw conclusions about their average daily tips.

Individual records are assumed to be independent normally distributed with population standard deviation

= 12.8. Sample summaries were found as

Mean = X

= 74.2 and Sample SD = s = 12

1. At signicance level= 5%; is there sucient evidence to conclude that the population average

is above 70?

Show critical value, test statistic, and state the rejection rule that explains your decision

2. At signicance level= 5%; is there evidence to conclude that the population average diers

from 70?

Show critical value, test statistic, and state the rejection rule that explains your decision

Problem 2 [10 points = 5 + 5]

A sample of n = 16 waiters was selected at random to draw conclusions about their average daily wages.

Individual records are assumed to be independent normally distributed with population standard deviation

= = 12:8: Sample summaries were found as

Mean = X

= 75.44 and Sample SD = s = 12

1. At signicance level= 5%; is there sucient evidence to conclude that the population average

is below 82?

Show critical value, test statistic, and state the rejection rule that explains your decision

2. At signicance level= 5%; is there evidence to conclude that the population average diers

from 82?

Show critical value, test statistic, and state the rejection rule that explains your decision

Problem 3 [15 points = 8 + 7]

A sample of n = 16 waiters was selected at random to draw conclusions about their average daily tips.

Individual records are assumed to be independent normally distributed with population standard deviation

= 12.8. Sample summaries were found as

Mean = X

= 74.2 and Sample SD = s = 12

1. Estimate the population average with condence C = 0.90:

Show critical value, margin of error, and condence limits

2. Estimate the population average with condence C = 0.99:

Show critical value, margin of error, and condence limits

Problem 4: [15 points = 5 + 5 + 5]

Managers at an assembly plant design a study aimed at estimating the average time needed for a production

line. They assume that individual records are normally distributed with the standard deviation = = 6:4

minutes. The goal is to ensure that the 90% condence interval has margin of error no higher than 1.5

minutes. Determine the smallest sample size needed and answer questions listed below.

1. What critical value are you going to use?

2. Show the formula and numerical value for the sample size needed.

3. Will a collection of 49 records be suffcient?

Problem 5: [15 points = 5 + 5 + 5]

A sample of n = 9 respondents was selected at random to analyze their weekly grocery expenses. Individual

records are assumed to be independent normally distributed with unknown average () and population

standard deviation = 24.00 Sample summaries were found as follows:

Sample Mean = X

= 164.20 and Sample SD = s = 27

1. At signicance level= 0.05; do you have sucient evidence that the population mean diers

from 0 = 150? State the rejection rule and show the test statistic value.

2. At signicance level= 0.05; is there sucient evidence that the population mean is below 0 =

150? State the rejection rule and show the test statistic value.

3. Estimate the population average with condence C = 0.90: Show margin of error and critical

value needed.

Assume that you are designing a study of commute time for students residing within 10 miles from campus.

Individual records are viewed as normally distributed with population standard deviation = = 7:5

minutes. These questions are all about the smallest sample size needed.

Answer all questions below showing critical value and formula for each of them.

1. You want to estimate the average commute time with the condence level C = 0:99 and ensure that

the interval will be no wider than 2.4 minutes.

2. Your objective is to estimate the average commute time at the condence C = 0:95 and its width

will not exceed 2.5 minutes.

3. You need to estimate the population average with condence C = 0:90 and ensure that the margin

of error will be no higher than 1.3 minutes.

Educators assume that individual scores in undergraduate courses are normally distributed with population

standard deviation = = 12: A sample of n = 9 students has shown the mean

X

= 73.4

Answering questions below, please show critical values, test statistic, and rejection rule. After that, state

your decision.

1. At signicance level of= 0:01; do you have evidence that the population average score is above

65?

2. At the same signicance level,= 0:01; do you have evidence that the population average is below

80?