Exam 12: Inference About a Population

Socialist Party Voters A pollster in Italy wants to challenge a claim that 5% of the registered voters in his country are Socialists;he thinks the percentage is lower than that.In a test of hypothesis,H₀: p = 0.05 vs.H₁: p < 0.05,his random sample of size 1,000 registered voters revealed that the number of Socialists was 40. -{Socialist Voters Narrative} Test the hypotheses at the 5% significance level.

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

New Mousetrap After a financial analysis,the general manager of a large company decided that if more than 8% of potential buyers of a new mousetrap purchase that product,the company would show a profit.In a preliminary survey of 500 potential buyers,56 people say that they will buy the mousetrap. -{New Mousetrap Narrative} State the appropriate null and alternative hypotheses.

A random sample of 25 observations is selected from a normally distributed population.The sample variance is 10.In the 95% confidence interval for the population variance,the upper limit is:

The sales manager of a large multinational corporation is concerned that some salespersons perform very well and others quite poorly.To help analyze the problem he draws a random sample of 20 salespersons,determines their commission incomes (in thousands of dollars),and calculates the following statistics: $\bar { x }$ = $37.2,and s = $7.8.Do these statistics provide sufficient evidence at the 5% significance level to conclude that the population variance exceeds $35 million²?

Tractor Dealership An accountant was performing an audit for a tractor dealership.An auditor wants to examine the monetary error made by the purchasing order department in the month of July.He decided to randomly sample 100 of the 925 purchase orders for the month of July,and found the amount of error in each one.The statistics for this sample were: $\bar { x }$ = $6.0 and s = $17.012. -{Tractor Dealership Narrative} Why should the finite population correction factor (FPCF)be used for this problem? Explain.

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

Under what condition(s)does the test statistic for p have an approximate normal distribution?

Healthy Foods In segmenting the frozen meal market in Spain,a food manufacturer uses health and diet consciousness as the segmentation variable.Four segments are developed and coded as follows: 1: Concermed about eating healthy faods 2: Concerned about weight 3: Concerned about health because of illnese 4: Unconcermed To distinguish between groups,surveys are conducted.On the basis of a questionnaire,people are categorized as belonging to one of these groups.A recent survey asked a random sample of 1,500 Spanish adults (20 and over)to complete the survey.Out of the 1,500 people surveyed,360 indicated that they are concerned about eating healthy foods.The most recent information available reveals that there are 40,256,000 Spanish adults who are 20 and over. -{Healthy Foods Narrative} What is the objective of this problem?

If we have some idea about the value of sample proportion $\hat { P }$ ,we use that value in determining the sample size needed to estimate the population proportion p.

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. -A left-tailed area in the chi-squared distribution equals 0.95.For 6 degrees of freedom the critical value equals 12.592.

Sinus Drug A company claims that 10% of the users of a certain sinus drug experience drowsiness.In clinical studies of this sinus drug,81 of the 900 subjects experienced drowsiness -{Sinus Drug Narrative} We want to test their claim and find out whether the actual percentage is not 10%.State the appropriate null and hypotheses.

Healthy Foods In segmenting the frozen meal market in Spain,a food manufacturer uses health and diet consciousness as the segmentation variable.Four segments are developed and coded as follows: 1: Concermed about eating healthy faods 2: Concerned about weight 3: Concerned about health because of illnese 4: Unconcermed To distinguish between groups,surveys are conducted.On the basis of a questionnaire,people are categorized as belonging to one of these groups.A recent survey asked a random sample of 1,500 Spanish adults (20 and over)to complete the survey.Out of the 1,500 people surveyed,360 indicated that they are concerned about eating healthy foods.The most recent information available reveals that there are 40,256,000 Spanish adults who are 20 and over. -{Healthy Foods Narrative} Interpret the confidence interval of p.

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:

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. -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.

The statistic s² is an unbiased and consistent estimator of $\sigma$ ².

The test statistic used to test hypotheses about the population variance has a(n)____________________ distribution with ____________________ degrees of freedom.

The width of a confidence interval estimate for a proportion will be:

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. -A condition of using the chi-squared test statistic for testing the variance is that the population random variable have a normal distribution.

Which of the following would be an appropriate null hypothesis?