Exam 7: Inferences Based on a Single Sample: 355 Tests of Hypotheses

Find the rejection region for the specified hypothesis test. -Consider a test of $\mathrm { H } _ { 0 } : \mu = 6$ . For the following case, give the rejection region for the test in terms of the $z$ -statistic: $\mathrm { H } _ { \mathrm { a } } : \mu \neq 6 , \alpha = 0.10$

A small private college is interested in determining the percentage of its students who live off campus and drive to class. Specifically, it was desired to determine if less than $20 \%$ of their current students live off campus and drive to class. The college decided to take a random sample of 108 of their current students to use in the analysis. Is the sample size of $n = 108$ large enough to use this inferential procedure?

A national organization has been working with utilities throughout the nation to find sites for large wind machines that generate electricity. Wind speeds must average more than 10 miles per hour (mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a particular site. Based on a sample of $n = 100$ wind speed recordings (taken at random intervals), the wind speed at the site averaged $\bar { x } = 9 \mathrm { mph }$ , with a standard deviation of $s = 3.0 \mathrm { mph }$ . To determine whether the site meets the organization's requirements, consider the test, $H _ { 0 } : \mu = 10$ vs. $H _ { \mathrm { a } } : \mu > 10$ , where $\mu$ is the true mean wind speed at the site and $\alpha = .05$ . Fill in the blanks. "A Type I error in the context of this problem is to conclude that the true mean wind speed at the site $10 \mathrm { mph }$ when it actually $10 \mathrm { mph } . "$

In a test of $H _ { 0 } : \mu = 12$ against $H _ { \mathrm { a } } : \mu > 12$ , a sample of $n = 75$ observations possessed mean $\bar { x }$ $= 13.1$ and standard deviation $s = 4.3$ . Find and interpret the $p$ -value for the test.

Suppose we wish to test $H _ { 0 } : \mu = 40 \mathrm { vs } . H _ { \mathrm { a } } : \mu > 40$ . What will result if we conclude that the mean is greater than 40 when its true value is really 47 ?

A random sample of $n = 12$ observations is selected from a normal population to test $H _ { 0 } : \mu$ $= 22.1$ against $H _ { \mathrm { a } } : \mu > 22.1$ at $\alpha = .05$ . Specify the rejection region.

A company claims that 9 out of 10 doctors (i.e., 90%) recommend its brand of cough syrup to their patients. To test this claim against the alternative that the actual proportion is less than 90%, a random sample of 100 doctors was chosen which resulted in 94 who indicate that they recommend this cough syrup. The test statistic in this problem is approximately:

The hypotheses for H0: $\mu = 65 \text { and } H _ { \mathrm { a } } : \mu > 65 \text { are tested at } \alpha = .05$ .05. Sketch the appropriate rejection region.

A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time spent on a delivery does not exceed 40 minutes. The owner has randomly selected 17 customers and delivered pizzas to their homes in order to test whether the mean delivery time actually exceeds 40 minutes. Suppose the p-value for the test was found to be .0293. State the correct conclusion.

It is desired to test $\mathrm { H } _ { 0 } : \mu = 50$ against $\mathrm { H } _ { \mathrm { A } } : \mu \neq 50$ using $\alpha = 0.10$ . The population in question is uniformly distributed with a standard deviation of 15 . A random sample of 49 will be drawn from this population. If $\mu$ is really equal to 45 , what is the power of the test?

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. - $z < - 1.28$

Data were collected from the sale of 25 properties by a local real estate agent. The following printout concentrated on the land value variable from the sampled properties. HYPOTHESIS: MEAN $X = x$ X = land - value SAMPLE MEAN OF X =51,611 SAMPLE VARIANCE OF X =273,643,254 SAMPLE SIZE OF X =25 x =47,168 MEANX- =4443 =1.34293 D.F. =24 P-VALUE =0.1918585 P -VALUE /2 =0.0959288 SD. ERROR =3308.43 What is the correct conclusion when testing a greater-than alternative hypothesis at $\alpha = .025$ ?

It is desired to test $H _ { 0 } : \mu = 55$ against $H _ { \mathrm { a } } : \mu < 55$ using $\alpha = .10$ . The population in question is uniformly distributed with a standard deviation of 15 . A random sample of 49 will be drawn from this population. If $\mu$ is really equal to 50 , what is the power of this test?

Find the rejection region for the specified hypothesis test. -A Type I error occurs when we accept a false null hypothesis.

A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time spent on a delivery does not exceed 36 minutes. The owner has randomly selected 21 customers and delivered pizzas to their homes in order to test whether the mean delivery time actually exceeds 36 minutes. What assumption is necessary for this test to be valid?

A random sample of 8 observations from an approximately normal distribution is shown below. 5 6 4 5 8 6 5 3 Find the observed level of significance for the test of $H _ { 0 } : \mu = 5$ against $H _ { \mathrm { a } } : \mu \neq 5$ . Interpret the result.

For the given binomial sample size and null-hypothesized value of p0, determine whether the sample size is large enough to use the normal approximation methodology to conduct a test of the null hypothesis $\mathrm { H } _ { 0 } : \mathrm { p } = \mathrm { P } 0$ -Consider the following printout. HYPOTHESIS: VARIANCE $X = x$ $X = \text { gpa }$ SAMPLE MEAN OF $X = 2.3827$ SAMPLE VARIANCE OF $X = .18000$ SAMPLE SIZE OF $X = 209$ HYPOTHESIZED VALUE $( x ) = 2.5$ VARIANCE $\mathrm { X } - \mathrm { x } = - .1173$ $z = - 3.99701$ Suppose we tested $H _ { \mathrm { a } } : \mu < 2.5$ . Find the appropriate rejection region if we used $\alpha = .05$ .

According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 556 bushels per acre. Twenty-five farmers who belong to a cooperative plant the soybeans in soil prepared as specified. Each uses a 40-acre plot and records the mean yield per acre. The mean and variance for the sample of 25 farms are $\bar { x } = 517 \text { and } s ^ { 2 } =$ 9580. Specify the null and alternative hypotheses used to determine if the mean yield for the soybeans is different than advertised.

Consider the following printout. HYPOTHESIS: MEAN X $= \mathrm { x }$ $X = \text { gpa }$ SAMPLE MEAN OF $X = 2.9397$ SAMPLE VARIANCE OF $X = 0.246823$ SAMPLE SIZE OF $X = 167$ HYPOTHESIZED VALUE $( \mathrm { x } ) = 3$ MEAN X-x =-0.0603 z =-1.56849 Suppose a two-tailed test is desired. Find the $p$ -value for the test.

In a test of $H _ { 0 } : \mu = 65$ against $H _ { \mathrm { a } } : \mu > 65$ , the sample data yielded the test statistic $z = 1.38$ . Find and interpret the $p$ -value for the test.