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Homework answers / question archive / Sanford-Brown College - MATH 2014 CHAPTER 12 SECTION 1: INFERENCE ABOUT A POPULATION 1)In order to determine the p-value associated with hypothesis testing about the population mean , it is necessary to know the value of the test statistic

Sanford-Brown College - MATH 2014 CHAPTER 12 SECTION 1: INFERENCE ABOUT A POPULATION 1)In order to determine the p-value associated with hypothesis testing about the population mean , it is necessary to know the value of the test statistic

Statistics

Sanford-Brown College - MATH 2014

CHAPTER 12 SECTION 1: INFERENCE ABOUT A POPULATION

1)In order to determine the p-value associated with hypothesis testing about the population mean , it is necessary to know the value of the test statistic.

 

          

 

 

 

     2.   In order to interpret the p-value associated with hypothesis testing about the population mean ?, it is necessary to know the value of the test statistic.

 

          

 

 

     3.   If a sample has 15 observations and a 95% confidence estimate for ? is needed, the appropriate value of t is 1.753.

 

          

 

 

     4.   If a sample has 18 observations and a 90% confidence estimate for ? is needed, the appropriate value of t is 1.740.

 

          

 

 

     5.   The statistic

 when the sampled population is normal is Student t-distributed with n degrees of freedom.

 

 

          

 

 

     6.   If the sampled population is nonnormal, the t-test of the population mean ? is still valid, provided that the condition is not extreme.

 

          

 

 

     7.   A race car driver tested his car for time from 0 to 60 mph, and in 20 tests obtained an average of 48.5 seconds with a standard deviation of 1.47 seconds. A 95% confidence interval for the 0 to 60 time is 45.2 seconds to 51.8 seconds.

 

          

 

 

     8.   In forming a 95% confidence interval for a population mean from a sample size of 20, the number of degrees of freedom from the t-distribution equals 20.

 

          

 

 

     9.   The t-distribution is used in a confidence interval for a mean when the actual standard error is not known.

 

          

 

 

   10.   The t-distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

 

          

 

 

   11.   The t-distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

 

          

 

 

   12.   The t-distribution assumes that the population is normally distributed.

 

          

 

 

   13.   In estimating the population mean with the population standard deviation unknown, if the sample size is 16, there are 8 degrees of freedom.

 

          

 

 

MULTIPLE CHOICE

 

   14.   For statistical inference about the mean of a single population when the population standard deviation is unknown, the degrees for freedom for the t-distribution equal n ? 1 because we lose one degree of freedom by using the:

a.

sample mean as an estimate of the population mean.

b.

sample standard deviation as an estimate of the population standard deviation.

c.

sample proportion as an estimate of the population proportion.

d.

sample size as an estimate of the population size.

 

 

          

 

 

   15.   Researchers determined that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Puffs users yielded the following data on the number of tissues used during a cold:

 = 52 and s = 22. Suppose the test statistic does not fall in the rejection region at ? = 0.05. Which of the following conclusions is correct?

 

a.

At ? = 0.05, we do not reject H0.

b.

At ? = 0.05, we reject H0.

c.

At ? = 0.05, we accept H0.

d.

Both a and c.

 

 

          

 

 

   16.   A robust estimator is one that is:

a.

unbiased and symmetrical about zero.

b.

consistent and is also mound-shaped.

c.

efficient and less spread out.

d.

not sensitive to moderate nonnormality.

 

 

          

 

 

   17.   A random sample of size 15 taken from a normally distributed population revealed a sample mean of 75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean would equal:

a.

77.77

b.

72.23

c.

88.85

d.

77.27

 

 

          

 

 

   18.   A major electronics store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results:

 = $50.50 and s2 = 400. A 95% confidence interval for the average amount the credit card customers spent on their first visit to the chain's new store in the mall is:

 

a.

$50.50 ? $9.09.

b.

$50.50 ? $10.12.

c.

$50.50 ? $11.08.

d.

None of these choices.

 

 

          

 

 

   19.   Researchers determine that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Puffs users yielded the following data on the number of tissues used during a cold:

 = 52 and s = 22. Using the sample information provided, the value of the test statistic is:

 

a.

t = (52 ? 60) / 22

b.

t = (52 ? 60) / (22 / 100)

c.

t = (52 ? 60) / (22 / 1002)

d.

t = (52 ? 60) / (22 / 10)

 

 

          

 

 

   20.   For a 99% confidence interval of the population mean based on a sample of n = 25 with s = 0.05, the critical value of t is:

a.

2.7969

b.

2.7874

c.

2.4922

d.

2.4851

 

 

          

 

 

   21.   Based on sample data, the 90% confidence interval limits for the population mean are LCL = 170.86 and UCL = 195.42. If the 10% level of significance were used in testing the hypotheses H0: ? = 201 vs. H1: ? ? 201, the null hypothesis:

a.

would be rejected.

b.

would be accepted.

c.

would fail to be rejected.

d.

would become H0: ? ? 201

 

 

          

 

 

   22.   The degrees of freedom for the test statistic for ? when ? is unknown is:

a.

1

b.

n

c.

n ? 1

d.

None of these choices.

 

 

          

 

 

   23.   Researchers determined that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Puffs users yielded the following data on the number of tissues used during a cold:

 = 52 and s = 22. Suppose the alternative we wanted to test was H1: ? < 60. The correct rejection region for ? = 0.05 is:

 

a.

reject H0 if t > 1.6604.

b.

reject H0 if t < ?1.6604.

c.

reject H0 if t > 1.9842 or Z < ?1.9842.

d.

reject H0 if t < ?1.9842.

 

 

          

 

 

COMPLETION

 

   24.   When the population standard deviation is ____________________ and the population is normal, the test statistic for testing hypotheses about ? is the t-distribution with n ? 1 degrees of freedom.

 

 

          

 

 

   25.   When the population standard deviation is unknown and the population is ____________________, the test statistic for testing hypotheses about ? is the t-distribution with n ? 1 degrees of freedom.

 

 

          

 

 

   26.   When the population standard deviation is unknown and the population is normal, the test statistic for testing hypotheses about ? is the ____________________-distribution with ____________________ degrees of freedom.

 

 

          

 

 

   27.   The t-test for a population mean is ____________________, meaning that if the population is nonnormal, the results of the test and confidence interval are still valid as long as the nonnormality is not extreme.

 

 

          

 

 

   28.   When a population is small, we must adjust the test statistic and interval estimator using the ____________________ population correction factor.

 

 

          

 

 

   29.   ____________________ populations allow us to use the confidence interval estimate of a mean to produce a confidence interval estimate of the population total.

 

 

          

 

 

   30.   The t-statistic has two variables: the sample ____________________ and the sample ____________________.

 

 

          

 

 

   31.   Because of the greater uncertainty, the t-statistic will display greater ____________________ than the z-statistic.

 

 

          

 

 

SHORT ANSWER

 

Single Mothers’ Ages

 

A random sample of 10 single mothers was drawn from a Obstetrics Clinic. Their ages are 22, 17, 27, 20, 23, 19, 24, 18, 19, and 24 years.

 

 

   32.   {Single Mothers' Ages Narrative} Estimate the population mean with 90% confidence.

 

 

          

 

 

   33.   {Single Mothers' Ages Narrative} Test to determine if we can infer at the 5% significance level that the population mean is not equal to 20.

 

 

          

 

 

   34.   {Single Mothers' Ages Narrative} What is the required condition of the techniques used in the previous questions? What graphical device can you use to check to see if that required condition is satisfied?

 

 

          

 

 

Concert Tickets

 

A simple random sample of 100 concert tickets was drawn from a normal population. The mean and standard deviation of the sample were $120 and $25, respectively.

 

 

   35.   {Concert Tickets Narrative} Test the hypothesis H0: ? = 125 vs. H1: ? ? 125 at the 10% significance level.

 

 

          

 

 

   36.   {Concert Tickets Narrative} Estimate the population mean with 90% confidence.

 

 

          

 

 

   37.   {Concert Tickets Narrative} Explain how to use the confidence interval to test the hypotheses at ? = 0.10.

 

 

          

 

 

Hourly Fees

 

A random sample of 15 hourly fees for car washers (including tips) was drawn from a normal population. The sample mean and sample standard deviation were

 = $14.9 and s = $6.75.

 

 

 

   38.   {Hourly Fees Narrative} Can we infer at the 5% significance level that the mean fee for car washers (including tips) is greater than 12?

 

 

          

 

 

   39.   {Hourly Fees Narrative} Can we infer at the 5% significance level that the population mean is greater than 12, assuming that you know the population standard deviation is equal to 6.75?

 

 

          

 

 

Tire Rotation

 

The manager of a service station is in the process of analyzing the number of times car owners rotate the tires on their cars. She believes that the average motorist rotates his or her car's tires less frequently than recommended by the owner's manual (two times per year). In a preliminary survey she asked 14 car owners how many times they rotated their cars' tires in the last 12 months. The results are 1, 1, 2, 0, 3, 3, 0, 1, 0, 1, 2, 3, 3, and 1.

 

 

   40.   {Tire Rotation Narrative} Does this data provide sufficient evidence at the 10% significance level to indicate that the manager is correct?

 

 

          

 

 

   41.   {Tire Rotation Narrative} What condition is required in order to analyze this data using a t-test?

 

 

          

 

 

   42.   The air pumps at service stations come equipped with a gauge to regulate the air pressure of tires. A mechanic believes that the gauges are in error by at least 3 pounds per square inch. To test his belief he takes a random example of 50 air pump gauges and determines the difference between the true pressure (as measured by an accurate measuring device) and the pressure shown on the air pump gauge. The mean and the standard deviation of the sample are

 = 3.4 and s = 1.2. Can the mechanic infer that he is correct at the 5% significance level? Assume tire pressures have a normal distribution.

 

 

 

          

 

 

   43.   A life insurance representative believes that the mean age of people who buy their first life insurance plan is less than 35. To test his belief he takes a random sample of 15 customers who have just purchased their first life insurance. Their ages are 42, 43,28, 34, 30, 36, 25, 29, 32, 33, 27, 30, 22, 37, and 40. There is not enough evidence to say the data are nonnormal. Can we conclude at the 1% significance level that the insurance representative is correct?

 

 

          

 

 

Mystic Pizza

 

Mystic Pizza in Mystic, Connecticut, advertises that they deliver your pizza within 15 minutes of placing an order or it is free. A sample of 25 customers is selected at random. The average delivery time in the sample was 13 minutes with a sample standard deviation of 4 minutes.

 

 

   44.   {Mystic Pizza Narrative} We want to know whether Mystic can make this claim or not. Test to determine if we can infer at the 5% significance level that the population mean delivery time is less than 15 minutes.

 

 

          

 

 

   45.   {Mystic Pizza Narrative} What is the required condition of the technique used in the previous question?

 

 

          

 

 

Energy Drink Consumption

 

A researcher at the University of Washington medical school believes that energy drink consumption may increase heart rate. Suppose it is known that heart rate (in beats per minute) is normally distributed with an average of 70 bpm for adults. A random sample of 25 adults was selected and it was found that their average heartbeat was 73 bpm after energy drink consumption, with a standard deviation of 7 bpm.

 

 

   46.   {Energy Drink Consumption Narrative} Formulate the null and alternative hypotheses.

 

 

          

 

 

   47.   {Energy Drink Consumption Narrative} Test the hypotheses in the previous question at the 10% significance level to determine if we can infer that energy drink consumption increases heart rate.

 

 

          

 

 

   48.   Employees in a large company are entitled to 15-minute water breaks. A random sample of the duration of water breaks for 10 employees was taken with the times shown as: 12, 16, 14, 18, 21, 17, 19, 15, 18, and 16. Assuming that the times are normally distributed, is there enough evidence at the 5% significance level to indicate that on average employees are taking longer water breaks than they are entitled to?

 

 

          

 

 

   49.   During a natural gas shortage, a gas company randomly sampled residential gas meters in order to monitor daily gas consumption. On a particular day, a sample of 100 meters showed a sample mean of 250 cubic feet and a sample standard deviation of 50 cubic feet. Provide a 90% confidence interval estimate of the mean gas consumption for the population.

 

 

          

 

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