Problem (bonus)Allen's hummingbird has been studied by zoologist Bill Alther. A small group of of 15 Allen's hummingbirds has been under study in Arizona. The average weight of these birds is ??¯= 3.15 grams. Based on previous Studies, we can assume that the weights of Allen's hummingbirds have a normal distribution , with ?? = 0.33 gram.
a) Find an 85% confidence interval for the average weights of Allen's hummingbirds in the studied region. What is the margin of error?
b) Give a brief interpretation of your results in the context of this problem.
c) Find the sample size necessary for an 85% confidence level with a maximal error of estimate ?? = 0.08 for the mean weights Allen's hummingbirds.
d) What will happen if we do part d) with 95% confidence level? Will the sample size increase or decrease?
Problem 1. A company claims that the average yearly salary of its workers is $29,800. The union believes that the average yearly salary is much less. A random sample of size 61 employees shows their average salary to be $29,400 with a sample standard deviation of $800. Use a 5% level of significance to test the company's claim.
a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?
b) What is the test statistic and its value? (Round your answer to three decimal places.)
c) Find (or estimate) the P-value. (Round your answer to four decimal places.)
d) Will you reject or fail to reject the null hypothesis? Explain how you made your decision
Problem 2. A match company sells matchboxes that are supposed to have an average of 40 matches per box with σ = 9. A random sample of 94 matchboxes made by this company gives an average number of matches per box equal to 43.1. Using a 1% level of significance, can you say that the average number of matches per box is more than 40?
Problem 3. The types of browse favored by deer are shown in the following table. Using binoculars, volunteers observed the feeding habits of a random sample of 320 deer.
| Type of Browse |
Plant Compositionin Study Area |
Observed Number of Deer Feeding on This Plant |
| Sage brush |
32% |
93 |
| Rabbit brush |
38.70% |
135 |
| Salt brush |
12% |
40 |
| Service berry |
9.30% |
32 |
| Other |
8% |
20 |
We want to test the claim that the natural distribution of browse fits the deer feeding pattern using a 5% level of significance.
a) State the null and alternate hypotheses.
(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.) What is the degree of freedom?
(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? Explain how you made your decision.
Problem 4. We have the following measurements (in minutes) about the time for 2 headache remedies to enter the bloodstream. Brand A: ??¯¯1¯ = 20.1; ??1 = 8.7; ??1 = 12. Brand B: ??¯¯2¯ = 11.21; ??2 = 7.5; ??2 = 8. Brand B claims to be faster. Write down the hypotheses for an appropriate statistical test, calculate the test statistic and the P-value. Make a decision
a) at the 1% level of significance
b) then at the 5% level of significance
Problem 5. The following table shows the Myers-Briggs personality preference for a random sample of 519 people in the listed professions. T refers to "thinking type" and F refers to "feeling type".
| Occupation |
T |
F |
Row Total |
| Clergy |
57 |
91 |
148 |
| M.D. |
77 |
82 |
159 |
| Lawyer |
118 |
94 |
212 |
| Column Total |
252 |
267 |
519 |
Use the chi-square test to determine if the listed occupations and personality preferences are independent.
Use 0.01 as a level of significance.