Exam 8: Hypothesis Testing

A fleet of rental cars - all the same make, model, and year - has a mean fuel efficiency of $24.1$ miles per gallon (mpg). A random sample of 40 cars are selected and the air filter of each is replaced with a new one. Let $\mu$ be the population mean fuel efficiency score that would occur if every car's air filter were replaced. The air filter change is deemed effective if $\mu > 24.1 \mathrm { mpg }$ . A test is made of $H _ { 0 } : \mu = 24.1$ versus $H _ { 1 } : \mu > 24.1$ . Assume that the air filter changes are effective. Which type of error is impossible?

The average magnesium concentration in ground water around Metro City is $\mu = 59.3$ parts per million (ppm) with a standard deviation of $\sigma = 18$ . The table below shows the ground-water magnesium concentrations $( \mathrm { ppm } )$ from random sites in a suburb located 15 miles away. At $\alpha = 0.10$ , can it be concluded that the average magnesium concentration in the suburb differs from $59.3 \mathrm { ppm }$ ? 43 62 56 32 36 116 27 9 47 111 115 96 27 55 79 59 63 43 36 44 102 54 32 64 93 78 8 105 110 84 124 125 104 104 87 a. State the hypotheses and identify the claim. b. Find the critical values. c. Compute the test statistic. d. Make a decision. e. Summarize the results.

At a certain university, the average attendance at basketball games has been 2825. Due to the dismal showing of the team this year, the attendance for the first 14 games has Averaged only 2515 with a standard deviation of 485. The athletic director claims that The attendance is the same as last year. What is the test value needed to evaluate the Claim?

According to Beautiful Bride magazine, the average age of a groom is now $26.2$ years. sample of 16 prospective grooms in Chicago revealed that their average age was $26.6$ years with a standard deviation of $5.3$ years. What is the test value for a $t$ test of the claim? A

If the null hypothesis is not rejected when it is false, a type II error has occurred.

Using Table G, find the $P$ -value interval for the $\chi ^ { 2 }$ test value. $\chi ^ { 2 } = 2.809 , n = 12 \text {, left-tailed }$

The following display from a TI-84 Plus calculator presents the results of a hypothesis test for a population proportion $p$ . Can $H _ { 0 }$ be rejected at the $0.05$ level?

A manufacturer claims that its televisions have an average lifetime of at least five years (60 months) with a population standard deviation of seven months. Eighty-one televisions were selected at random, and the average lifetime was found to be 59 months. With $\alpha = 0.025$ , is the manufacturer's claim supported?

At a water bottling facility, a technician is testing a bottle filling machine that is supposed to deliver 500 milliliters of water. The technician dispenses 49 samples of water and determines the volume of each sample. The 49 samples have a mean volume of $\bar { x } = 501.2 \mathrm {~mL}$ . The machine is out of calibration if the mean volume differs from $500 \mathrm {~mL}$ . The technician wants to perform a hypothesis test to determine whether the machine is ou calibration. State the appropriate null and alternate hypotheses.

The Eagle Ridge Contractors Association claims the average price of a home in their subdivision is $125,150 with a standard deviation of $7,350. A sample of 36 homes for Sale in this subdivision had an average selling price of $123,550. The Eagle Ridge Home Owners Association is interested in knowing if the costs of homes for sale in this Subdivision are actually lower than claimed? Compute the test value.

In a(n) test, the null hypothesis should be rejected when the test value is in either of the two critical regions.

At a certain university, 16% of students fail general chemistry on their first attempt. Professor Brown teaches at this university and believes that the rate of first-time failure in his general chemistry classes is 33%. He samples 96 students from last semester who were first-time enrollees in general chemistry and finds that 15 of them failed his course. i). State the appropriate null and alternate hypotheses. ii). Compute the test statistic $\mathrm { z }$ . iii). Using $\alpha = 0.05$ , can you conclude that the percentage of failures differs from $33 \%$ ?

The Golden Comet is a hybrid chicken that is prized for its high egg production rate and gentle disposition. According to recent studies, the mean rate of egg production for 1-year-old Golden Comets is $5.0$ eggs/week. Sarah has 39 1-year-old hens that are fed exclusively on natural scratch feed: insects, seeds, and plants that the hens obtain as they range freely around the farm. Her hens exhibit a mean egg-laying rate of $5.2$ eggs/day. Sarah wants to determine whether the mean laying rate $\mu$ for her hens is higher tha the mean rate for all Golden Comets. Assume the population standard deviation to be $\sigma = 1.1 \mathrm { eggs } /$ day. i. State the appropriate null and alternate hypotheses. ii. Compute the value of the test statistic. iii. State a conclusion. Use the $\alpha = 0.01$ level of significance.

The following output from MINITAB presents the results of a hypothesis test for a population mean $\mu$ . Test of $m u = 37$ vs. not $= 37$ Mean StDev SE Mean 95\% CI T P 44 34.14 10.214809 1.53994 (31.034387,37.245613) -1.857215 0.0701 What are the null and alternate hypotheses?

If the null hypothesis $H _ { 0 } : \mu = 11.0$ is not rejected at $\alpha = 0.05$ when a mean of $8.0$ is obtained from a random sample, one could also say that

The numerical value obtained from a statistical test is called the .

A recent survey indicated that the average amount spent for breakfast by business managers was $\$ 7.58$ with a standard deviation of $\$ 0.42$ . It was felt that breakfasts on the West Coast were higher than $\$ 7.58$ . A sample of 81 business managers on the West Coast had an average breakfast cost of $\$ 7.65$ . At $\alpha = 0.05$ , what is the critical value?

A garden supplier claims that its new variety of giant tomato produces fruit with an mean weight of 42 ounces. A test is made of $H _ { 0 } : \mu = 42$ versus $H _ { 1 } : \mu \neq 42$ . The null hypothesis is rejected. State the appropriate conclusion.