# Days to recover from a cold; sound frequencies

A random selection of volunteers at a research institute have been exposed to a typical cold virus. After they started to have cold symptoms, 15 of them were given multivitamin tablets daily which contain 1 gram of vitamin C and various other vitamins and minerals. The remaining 15 volunteers were given tablets only containing 4 grams of vitamin C. For each individual, the length of time taken to recover from cold is recorded. At the end of the experiment the following data are obtained:

Days to recover from cold

Treated with multivitamin 7.4, 4.6, 5.9, 8.1, 2.1, 7.6, 6.8, 5.3, 8.3, 6.9, 4.0, 6.4, 7.4, 2.6, 4.8.

Treated with Vitamin C 6.4, 3.5, 5.1, 4.7, 4.3, 4.2, 3.1, 4.1, 5.4, 2.2, 5.0, 3.9, 3.6, 2.4, 4.0.

It is known that the population standard deviation of recovery time from cold is 18 days when treated with multivitamin, and the population standard deviation of recovery time from cold is1.5 days when treated with

vitamin C tablets. It is also known that both populations are approximately normally distributed. The researchers claim that the mean recovery time, μ1 ,of the patients treated with multivitamin is less than or equal to the mean

recovery time , μ 2, of the patients who are treated with vitamin C tablets. At the 0.05 level of significance, is there enough evidence to reject this claim? Perform a one-tailed test. Then answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the

questions.

1. The null hypothesis:

H0

:

2.The alternative hypothesis:

H1

:

3. The type of test statistic:

a. Z

b. t

c. Chi-Square

d. F

4. The value of the test statistic:

(Round to at least three decimal places.)

5. The critical value at the 0.05 level of significance:

(Round to at least three decimal places.)

6. Can we reject the researchers' claim that the mean recovery time when treated with multivitamin is less than or equal to the mean recovery time when treated with vitamin C only?

a. Yes b. No

A random selection of volunteers at a research institute have been exposed to a typical cold virus. After they started to have cold symptoms, 10 of them were given multivitamin tablets daily which contain 3 grams of vitamin C and various other vitamins and minerals. The remaining 10 volunteers were given placebo tablets. For each individual, the length of time taken to recover from cold is recorded. At the end of the experiment following data are obtained:

Days to recover from cold

Treated with multivitamin 5.7, 3.2, 6.2, 5.7, 3.5, 2.5, 5.0, 4.2, 7.7, 3.8.

Treated with placebo 3.6, 6.1, 7.5, 4.9, 5.2, 2.6, 4.3, 2.8, 4.6, 3.9.

It is known that the population standard deviation of recovery time from cold is 1.8 days when treated with

multivitamin, and the population standard deviation of recovery time from cold is 1.5 days when treated with

placebo tablets. It is also known that both populations are approximately normally distributed. The researchers

claim that the mean recovery time, population (u1) , of the patients treated with multivitamin is not equal to the mean recovery time , population (u2), of the patients who are treated with placebo tablets. At the 0.1 level of significance, is there enough evidence to support this claim? Perform a two-tailed test. Then answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the

questions.

1. The null hypothesis:

H0

2. The alternative hypothesis:

H1

3. The type of test statistic:

a. Z

b. t

c. Chi-Square

d. 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.Can we support the researchers' claim that the mean recovery time when treated with multivitamin is not

equal to the mean recovery time when treated with placebo?

a. Yes b. No

3. Much is still to be learned about the relationship between sound frequency and loudness. One way to study the

relationship between sound frequency and loudness is to have listeners perform loudness judgments for tones of

different frequencies. For each listener, the output of these judgments is a number, measured in sones, that gives

the loudness of the tone relative to the loudness of a reference tone. Suppose that you have in front of you data from an experimental study in which listeners were asked to perform

such loudness judgments for tones of various intensities and frequencies. The listeners were divided into

non-overlapping groups according to their hearing ability ("normal, unaided hearing," "some hearing loss at certain

frequencies," "normal, aided hearing," etc.). The data give the sone measurements for each listener for a 50 dB

SPL, 500 -Hz tone. You perform a one-way, independent-samples ANOVA test of the hypothesis that the mean sone measurement are equal for the different populations of listeners represented in the study. This ANOVA test is summarized in the ANOVA table below. Fill in the missing value of this ANOVA table (round your answer to at least two decimal places), and then answer the questions below the table.

Source of Variation Degrees of

Freedom Sum of Squares Mean Square F statistic

Treatments

(Between Groups) 5 1.73 0.35

Error

(Within Groups) 90 19.65 0.22

Total 95 21.38

1. How many groups of listeners were tested in the experiment?

3. Using the 0.01 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. Using the 0.01 level of significance, do you conclude that there are differences in the mean sone values for this tone for the three populations of listeners? (a) Yes (b) No

4. Jointsoft is a great over-the-counter arthritis medication, but who will ever know about it? Unfortunately, many people with arthritis tend to be elderly and rather immobile, so advertisers of arthritis medications face limitations in ways to get their messages across. Currently, their best modes of advertisement are commercials on daytime TV, advertisements in select magazines, fliers in convalescent homes, and (believe it or not) advertisements on certain Web pages. Marketing managers for Jointsoft are investigating these four modes of advertisement in four small communities(with a different mode of advertisement in each community). The marketing managers have selected 36 days at random and are looking at the daily sales (in dollars) in each of the communities on each of these days. Here is what they have to work with:

Suppose that the marketing managers perform a one-way, independent-samples ANOVA test to decide if there are differences in the mean daily sales arising from the four modes of advertisement. (So, they're assuming that the only difference among the four communities is the mode of advertisement used in it.) Such a test uses the statistic

Variation between the samples.

F = Variation within the samples

For the information in the chart above,

1. Give the numerator degrees of freedom of this F statistic.

2. Give the denominator degrees of freedom of this F statistic.

3.Using the 0.01 level of significance, can the marketing managers conclude that the mean daily sales arising from at least one of the modes of advertisement differs from the others?

(a) Yes (b) No

5. Bivariate data for the quantitative variables X and Y are given in the table below. These data are plotted in the

scatter plot shown next to the table. In the scatter plot, sketch an approximation of the least-squares regression line for the data.

6. Below are four bivariate data sets and the scatter plot for each. (Note that each scatter plot is displayed on the

same scale.) Each data set is made up of sample values drawn from a population.

Figure 1

Figure 2

Figure 3

Figure 4

Answer the following questions. The same response may be the correct answer for more than one question.

1. Which data set indicates a perfect positive linear relationship between its two variables?

a. the x, y data set

b. the u, v data set

c. the w, t data set

d. the m, n data set

e. none of the data sets

2. Which data set has an apparent negative, but not perfect, linear relationship between its two variables?

a. the x, y data set

b. the u, v data set

c. the w, t data set

d. the m, n data set

e. none of the data sets

3. In which data set is there evidence of a strong nonlinear relationship between the two variables?

a. the x, y data set

b. the u, v data set

c. the w, t data set

d. the m, n data set

e. none of the data sets

4. Which data set indicates the strongest negative linear relationship between its two variables?

a. the x, y data set

b. the u, v data set

c. the w, t data set

d. the m, n data set

7. An advertising firm wishes to demonstrate to its clients the effectiveness of the advertising campaigns it has conducted. The following bivariate data on fifteen recent campaigns, including the cost of each campaign (in millions of dollars) and the resulting percentage increase in sales following the campaign, were presented by the firm. Based on these data, we would compute the least-squares regression line to be , Y=6.13+0.20x with x representing campaign cost and y representing the resulting percentage increase in sales. (This line is shown in Figure 1, along with a scatter plot of the data.)

Answer the following:

1. Fill in the blank: For these data, values for percentage increase in sales that are greater than the mean of the values for percentage increase in sales tend to be paired with values for campaign cost that are _____ the mean of the values for campaign cost.

a. greater than b. less than

2. Fill in the blank: According to the regression equation, for an increase of one million dollars in advertising campaign cost, there is a corresponding _____ of 0.20 percent in sales.

a. increase b. decrease

3. From the regression equation, what is the predicted percentage increase in sales when the advertising campaign cost is 1.43 million dollars? (Round your answer to at least two decimal places.)

4..What was the observed percentage increase in sales when the advertising campaign cost was 1.43 million dollars?

.

8. A financial analyst is examining the relationship between stock prices and earnings per share. She chooses fifteen,

publicly traded companies at random and records for each the company's current stock price and the company's

earnings per share reported for the past 12 months. Her data are given below, with x denoting the earnings per share from the previous year and y denoting the current stock price (both in dollars). A scatter plot of her data is shown in Figure 1.

The value of the sample correlation coefficient y for these data is approximately 0.879.

Answer the following. Carry your intermediate computations to at least four decimal places, and round your answers

as specified below.

What is the value of the slope of the least-squares regression line for these data? Round your answer to at least three decimal places.

What is the value of the y-intercept of the least-squares regression line for these data? Round your answer to at least three decimal places.

9. Using 23 observations on each variable, a computer program generated the following multiple regression model:

If the standard errors of the coefficients of the independent variables are, respectively, 1.22 and 1.27

can you conclude that the independent variable x2 is needed in the regression model?

Let B1 and B2 denote the coefficients of the 2 variables in this model, and use a two-sided hypothesis test and significance level of 0.01 to determine your answer.

Carry your intermediate computations to at least three decimal places and round your answers as specified.

1.The null hypothesis:

H0

2.The alternative hypothesis:

H1

3. The type of test statistic:

a. Z

b. t

c. Chi-Square

d. F

4. The value of the test statistic:

(Round to at least two decimal places.)

5. The two critical values at the 0.01 level of significance:

(Round to at least two decimal places.)

1. Can you conclude that the independent variable x2 is needed in the regression model?

a. Yes b. No

Two popular drugs used for the treatment of depression are Resithan and Exemor. A random sample of 437depressed individuals is selected and treated with Resithan, and 143 find relief from their depression. A random sample of 567 depressed individuals is independently selected from the first sample and treated with Exemor, And 191 find relief from their depression. Can we conclude, at the 0.1 level of significance, that the proportion p1 of depressed individuals taking Resithan who find relief from depression is less than the proportion p2 of all

depressed individuals taking Exemor who find relief from depression?

Perform a one-tailed test. Then answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the

questions.

1. The null hypothesis:

H0

2. The alternative hypothesis:

H1

3. The type of test statistic:

a. Z

b. t

c. Chi-Square

d. F

4.The value of the test statistic:

(Round to at least three decimal places.)

5. The critical value at the 0.1 level of significance:

(Round to at least three decimal places.)

6.Can we conclude that the proportion of depressed individuals taking Resithan who find relief is less than the proportion taking Exemor who find relief?

a. Yes b. No

11. A coin-operated drink machine was designed to discharge a mean of 9 ounces of coffee per cup. Suppose that

we want to carry out a hypothesis test to see if the true mean discharge differs from 9. State the null hypothesis H0

and the alternative hypothesis H1 that we would use for this test.

H0

H1

12. The following time series data represent the yearly amounts spent on advertising (in millions of dollars) by a large toy company:

7.5, 9.9, 8.5, 9.5, 11.6, 10.8, 10.4, 13.9, 13.9, 13.4

This series of data begins in year 1993 (i.e., time period t = 1 corresponds to 1993 ). Using regression

analysis, a linear trend line of the form was fit to the data. Using this information, generate a forecast for the total yearly amount of money that will be spent on advertising in 2010.

We want to predict the selling price of a house in Newburg Park, Florida, based on the distance the house lies from the beach. Suppose that we're given the data in the table below. These data detail the distance from the beach (x, in miles) and the selling price ( y, in thousands of dollars) for each of a sample of fifteen homes sold in Newburg Park in the past year. The data are plotted in the scatter plot in Figure 1. Also given are the products of the distances from the beach and house prices for each of the fifteen houses. (These products, written in the column labelled "xy ," may aid in calculations.)

Figure 1

Answer the following. Carry your intermediate computations to at least four decimal places, and round your answer as specified below.

What is the value of the sample correlation coefficient for these data? Round your answer to at least three decimal

places.

#### Solution Preview

(Answers in in this Orange color)

A random selection of volunteers at a research institute have been exposed to a typical cold virus. After they started to have cold symptoms, 15 of them were given multivitamin tablets daily which contain 1 gram of vitamin C and various other vitamins and minerals. The remaining 15 volunteers were given tablets only containing 4 grams of vitamin C. For each individual, the length of time taken to recover from cold is recorded. At the end of the experiment the following data are obtained:

Days to recover from cold

Treated with multivitamin 7.4, 4.6, 5.9, 8.1, 2.1, 7.6, 6.8, 5.3, 8.3, 6.9, 4.0, 6.4, 7.4, 2.6, 4.8.

Treated with Vitamin C 6.4, 3.5, 5.1, 4.7, 4.3, 4.2, 3.1, 4.1, 5.4, 2.2, 5.0, 3.9, 3.6, 2.4, 4.0.

It is known that the population standard deviation of recovery time from cold is 18 days when treated with multivitamin, and the population standard deviation of recovery time from cold is 1.5 days when treated with

vitamin C tablets. It is also known that both populations are approximately normally distributed. The researchers claim that the mean recovery time, μ1 ,of the patients treated with multivitamin is less than or equal to the mean

recovery time , μ 2, of the patients who are treated with vitamin C tablets. At the 0.05 level of significance, is there enough evidence to reject this claim? Perform a one-tailed test. Then answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the

questions.

1. The null hypothesis:

H0: μ1- μ 2≤0(because it is a claim)

:

2.The alternative hypothesis:

H1: μ1- μ 2>0

:

3. The type of test statistic:

a. Z (since pop means are known)

b. t

c. Chi-Square

d. F

4. The value of the test statistic:

(Round to at least three decimal places.)

Z=0.063548 (I used Excel)

5. The critical value at the 0.05 level of significance:

(Round to at least three decimal places.)

1.645

6. Can we reject the researchers' claim that the mean recovery time when treated with multivitamin is less than or equal to the mean recovery time when treated with vitamin C only?

a. Yes b. No p-value =0.06>0.05

Excel Output Here:

z-Test: Two Sample for Means

MultiVitamin Vit C

Mean 5.88 4.135714286

Known Variance 18 1.5

Observations 15 14

Hypothesized Mean Difference 0

z 1.525654158

P(Z<=z) one-tail 0.063548008

z Critical one-tail 1.644853627

P(Z<=z) two-tail 0.127096017

z Critical two-tail 1.959963985

A random selection of volunteers at a research institute have been exposed to a typical cold virus. After they started to have cold symptoms, 10 of them were given multivitamin tablets daily which contain 3 grams of vitamin C and various other vitamins and minerals. The remaining 10 volunteers were given placebo tablets. For each individual, the length of time taken to recover from cold is recorded. At the end of the experiment following data are obtained:

Days to recover from cold

Treated with multivitamin 5.7, 3.2, 6.2, 5.7, 3.5, 2.5, 5.0, 4.2, 7.7, 3.8.

Treated with placebo 3.6, 6.1, 7.5, 4.9, 5.2, 2.6, 4.3, 2.8, 4.6, 3.9.

It is known that the population standard deviation of recovery time from cold is 1.8 days when treated with

multivitamin, and the population standard deviation of recovery time from cold is 1.5 days when treated with

placebo tablets. It is also known that both populations are approximately normally distributed. The researchers

claim that the mean recovery time, population (u1) , of the patients treated with multivitamin is not equal to the mean recovery time , population (u2), of the patients who are treated with placebo tablets. At the 0.1 level of significance, is there enough evidence to support this claim? Perform a two-tailed test. Then answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the

questions.

1. The null hypothesis:

H0: μ1- μ 2=0

2. The alternative hypothesis:

H1: μ1- μ2≠0

3. The type of test statistic:

a. Z

b. t

c. Chi-Square

d. F

4.The value of the test statistic:

(Round to at least three decimal ...

#### Solution Summary

The number of days to recover from a cold for the sound frequencies are examined.