Exam 10: Hypothesis Tests Involving a Sample Mean or Proportion

Consider testing the hypotheses H₀: $\mu$ = 50 vs.H₁: $\mu \neq$ 50. If n = 64, $\bar { x }$ = 53.5,and $\text { ? }$ = 10,then the value of the test statistic is:

A filling machine is designed to fill soda bottles with 16 ounces of soda.The distribution for the weight of the bottles is normal.Twenty bottles are selected and weighed.The sample mean is 15.3 ounces and sample standard deviation is 1.5 ounces.Develop a 90% confidence interval for this sample.What can we conclude about this filling process with $\alpha$ = 0.10?

If we do not reject the null hypothesis,we conclude that:

In testing the hypotheses H₀: $\pi$ = 0.40,H₁: $\pi$ > 0.40 at the 5% significance level,if the sample proportion is .45,and the standard error of the sample proportion is .035,the appropriate conclusion would be:

Calculate the probability of a Type II error for the following test of hypothesis H₀: $\mu$ = 50 versus H₁: $\mu$ > 50 given that $\mu$ = 55, $\alpha$ =.05, $\text { σ }$ = 10,and n = 16.

Table 1 Based on the sample data,a confidence interval has been constructed such that we have 95% confidence that the population mean is between 75 and 105. -What conclusion would be reached at the $\alpha$ = .05 level in testing H₀: $\mu$ = 65 versus H₁: $\mu \neq$ 65?

In a criminal trial where the null hypothesis states that the defendant is innocent,a Type I error is made when:

Local Restaurant A local restaurant claimed in an advertisement that at least 80% of their customers rate their establishment as "Excellent".An independent survey company asked 200 customers their opinion and from this random sample,168 customers rated the establishment "Excellent".Use $\alpha$ = 0.05. -H₀: $\mu \leq \mu _ { 0 }$ ,H₁: $\mu > \mu _ { 0 }$ ; n > 30 and $\alpha$ = 0.01 State the appropriate decision rule.

Restaurant A restaurant claims that the average gratuity per table is $15.00.To test this claim,a sample of 18 tables was selected which had an average gratuity of $13.75 with a standard deviation of $2.60.Assume the population of gratuities is normally distributed and use $\alpha$ = 0.10. -State the null and alternative hypotheses.

Suppose the underlying distribution is known to be normal,or that the sample size is at least thirty (30).The t-test is the appropriate test statistic when the sample standard deviation is used to estimate the population standard deviation.

Merit In order to receive a merit raise a computer clerk must average less than 15 minutes per order form processed.An across-the-board raise was given if the average is 15 minutes or more.The supervisor randomly selects 36 order forms and carefully times the length of time required to enter all of the data on each form.The significance level = 0.02.The average time to enter the 36 order forms is 14.36 minutes,and the population standard deviation is 3.03 minutes with a normal distribution.The test question is: Should a merit raise be given? -State the null and alternative hypotheses.

A non-directional claim or assertion states that a population parameter is equal to some quantity.

Mr.Perfection Mr.Perfection,the owner of a small unique restaurant called the Village Feed Store,is concerned with the waiting time for patrons on Sunday morning - the busiest day of the week.He is fearful that,though his establishment is extremely popular,excess waiting time will drive customers to Kenny's,a restaurant down the street.The hostess insists that customers do not have to wait more than an average of 10 minutes before being seated,but Mr.Perfection feels that the waiting time is longer.He takes a random sample of 100 customers over several Sundays.Assume that the population standard deviation $\text {σ}$ = 12 minutes,and that the significance level $\alpha$ = 0.05. -What is the probability of making a Type II error if the true mean is 14?

The power of a test is the probability that a true null hypothesis will be rejected.

For the critical z,identify the sample proportion that this z-value represents. p = ____________________

A spouse stated that the average amount of money spent on Christmas gifts for immediate family members is above $1200.The correct set of hypotheses is:

A production engineer is checking the volume in cans of paint as the cans leave the filling line.The machine is set to fill the cans with 4 liters.However,the engineer wishes to determine if the cans are being filled with less than 4 liters.A sample of 36 is taken.The sample shows a mean of 3.97 and a standard deviation of 0.14.Conduct the appropriate test at the 0.01level of significance. Appropriate test: _____________________ Test statistic = ____________________ Conclusion: ____________________ Interpretation: ____________________________________________________

What is a null hypothesis?

Soft drink A soft drink dispenser is set to automatically dispense drinks and ice in 8 oz.cups.The machine is set to fill the cups with 7 1/4 oz.so that the liquid will not run over the top of the cup.The company is getting complaints that the machine is filling the cups too full and spilling over.When the machine is operating properly,only 1% of the cups will not be filled properly.The production manager conducted a one-tail hypothesis test at the 0.01 level to check the machine.A sample of 50 fills was measured. What is the critical value of z?

Life span Research has claimed that the average life span of a male mosquito is 15 days.To test this claim,the life spans of 20 male mosquitos were observed to have an average of 16.5 days and a standard deviation of 2.2 days.Assume that the life spans are normally distributed and set $\alpha$ = 0.05. -State your conclusion.