# Hypothesis testing problems

Question #1 / 9

The mean SAT score in mathematics is . The founders of a nationwide SAT preparation course claim that graduates of the course score higher, on average, than the national mean. Suppose that the founders of the course want to carry out a hypothesis test to see if their claim has merit. State the null hypothesis and the alternative hypothesis that they would use.

1 H o:

2 H 1:

Question #2 / 9

The breaking strengths of cables produced by a certain manufacturer have a mean, , of pounds, and a standard deviation of pounds. It is claimed that an improvement in the manufacturing process has increased the mean breaking strength. To evaluate this claim, newly manufactured cables are randomly chosen and tested, and their mean breaking strength is found to be pounds. Can we support, at the level of significance, the claim that the mean breaking strength has increased? (Assume that the standard deviation has not changed.)

Perform a one-tailed test. Then fill in the table below.

Carry your intermediate computations to at least three decimal places, and round your responses as specified in the table. (If necessary, consult a list of formulas.)

1 the null hypothesis H o:

2 the alternative hypothesis H 1:

3 the type of test statistic: Z t chi-square F

4 the value of the test statistic: (Round to at least three decimal places).

5 The critical value at the 0.01 level of significance: (Round to at least three decimal places).

6 Can we support the claim that the mean breaking strength has increased? Yes No

Question #3 / 9

Heights were measured for a random sample of plants grown while being treated with a particular nutrient. The sample mean and sample standard deviation of those height measurements were centimeters and centimeters, respectively.

Assume that the population of heights of treated plants is normally distributed with mean . Based on the sample, can it be concluded that is different from centimeters? Use the level of significance.

Perform a two-tailed test. Then fill in the table below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

1 the null hypothesis H o:

2 the alternative hypothesis H 1:

3 the type of test statistic: Z t chi-square F

4 the value of the test statistic: (Round to at least three decimal places).

5 The p-value: (Round to at least three decimal places).

6 At the 0.05 level of significance, can it be concluded that the population mean height of treated plants is different from 36 centimeters? Yes No

Question #4 / 9

A manager at LLD Records is investigating the company's market research techniques. She learns that much of the market research of college students is done during promotions on college campuses. She also learns that there are other methods of performing market research (for instance, over the phone, in a mall, etc.). In all cases, for each new CD thta LLD Records releases, the company solicits an "intent-to-purchase" score from the student, with being the lowest score ("no intent to purchase") and being the highest score ("full intent to purchase").

The manager finds some information on a soon-to-be-released CD. The information details the intent-to-purchase scores from each of several groups of college students, with each group being questioned via a different method. Based on this information, the manager is able to perform a one-way, independent-samples ANOVA test of the hypothesis that the mean intent-to-purchase score for this CD is the same no matter the method of score collection. This test is summarized in the ANOVA table below.

Fill in the missing entry in the ANOVA table (round your answer to at least two decimal places), and then answer the questions below.

Source of variation Degrees of freedom Sum of squares Mean Square F statistic

1 Treatments (Between Groups) 2 290.7 145.4 

Error (Within Groups) 72 6540 90.8

Total 74 6830.7

1 How many intent-to purchase scores were examined in all? 

2 For the ANOVA test, it is assumed that the population variances of intent-to-purchase scores are the same no matter the method of score collection. What is an unbiased estimate of this common population variance based on the sample variances? 

3 Using the 0.10 level of significance, what is the critical value of the F statistic for the ANOVA test? Round your answer to at least two decimal places. 

4 Based on this ANOVA, can we conclude that there are differences in the mean intent-to-purchase scores (for this CD) among the different methods of collection? Use the 0.10 level of significance. Yes No

Question #5 / 9

Cris Turlock owns and manages a small business in San Francisco, California. The business provides breakfast and brunch food, via carts parked along sidewalks, to people in the business district of the city.

Being an experienced businessperson, Cris provides incentives for the four salespeople operating the food carts. This year, she plans to offer monetary bonuses to her salespeople based on their individual mean daily sales. Below is a chart giving a summary of the information that Cris has to work with. (In the chart, a "sample" is a collection of daily sales figures, in dollars, from this past year for a particular salesperson.)

Groups Sample Size Sample Mean Sample Variance

Salesperson 1 132 218.8 2461.9

Salesperson 2 92 222.3 1899.5

Salesperson 3 116 209.4 3017.9

Salesperson 4 129 212.2 2515.7

Cris' first step is to decide if there are any significant differences in the mean daily sales of her salespeople. (If there are no significant differences, she'll split the bonus equally among the four of them.) To make this decision, Cris will do a one-way, independent-samples ANOVA test of equality of the population means, which uses the statistic

Variation between the samples .

Variation within the samples

For these samples, .

1 Give the numerator degrees of freedom of this F statistic

2 Give the denominator degrees of freedom of this F statistics

3 Can we conclude, using the 0.05 level of significance, that at least one of the salespeople's mean daily sales is significantly different from that of the others? Yes No

Question #6 / 9

At LLD Records, some of the market research of college students is done during promotions on college campuses, while other market research of college students is done through anonymous mail, phone, internet, and record store questionnaires. In all cases, for each new CD the company solicits an "intent-to-purchase" score from the student, with being the lowest score ("no intent to purchase") and being the highest score ("full intent to purchase").

The manager finds the following information for intent-to-purchase scores for a soon-to-be-released CD:

Groups Sample Size Sample Mean Sample Variance

On Campus 25 64.4 127.2

By mail 25 68.7 96.5

By phone 25 62.9 40.5

By internet 25 67.0 98.4

In a store 25 62.8 219.6

The manager's next step is to conduct a one-way, independent-samples ANOVA test to decide if there is a difference in the mean intent-to-purchase score for this CD depending on the method of collecting the scores.

Answer the following, carrying your intermediate computations to at least three decimal places and rounding your responses to at least one decimal place.

1 What is the value of the mean square for error (the "within groups" mean square) that would be reported in the ANOVA test?

2 What is the value of the mean square for treatments (the"between groups" mean square) that would be reported in the ANOVA test?

Question #7 / 9

The Ellington Dukes are a minor-league baseball team in Ellington, Georgia. As with most other minor-league teams, the Dukes rely heavily on promotions to bring fans to the ballpark. These promotions are typically aimed at fans of specific ages. The management for the Dukes has planned its current promotional schedule according to the following estimates: of fans attending Dukes games are ages to , are ages to , are ages to , are ages to , and are over .

A statistical consulting firm for the Dukes surveyed a random sample of fans attending Dukes games in order to see if these estimates are accurate. The observed frequencies in the sample of for each of the age categories are given in the top row of numbers in Table 1 below. The second row of numbers contains the frequencies expected for a random sample of fans if the Dukes' management's estimates are accurate. The bottom row of numbers in Table 1 contains the values

= (Observed frequency - Expected frequency)2

Expected frequency

for each of the age categories.

Fill in the missing values of Table 1. Then, using the level of significance, perform a test of the hypothesis that the management's estimates are accurate. Then complete Table 2.

Round your responses for the expected frequencies in Table 1 to at least two decimal places. Round your responses in Table 1 to at least three decimal places. Round your responses in Table 2 as specified.

Table 1: Information about the Sample

Age Group

0 to 12 13 to 18 19 to 35 36 to 55 Over 55

Observed freq 35 23 51 39 32

1 Expected freq 18 45 36

2

1.389 0.8 0.444

Table 2: Summary of the Hypothesis Test

1 the type of test statistic: Z t chi-square F

2 the value of the test statistic:

(Round the answer to at least two decimal places).

3 The critical value for a test at the 0.10 level of significance: (Round your answer to at least two decimal places).

4 Can we conclude that the management's original estimates for the age distribution of fans attending Dukes games are inaccurate? Use the 0.10 level of significance. Yes No

#### Solution Summary

Step by step method for computing test statistic for hypothesis testing, ANOVA and regression are given in this answer.