Tests of Hypotheses Based on a Single Sample

Abstract

A parameter can be estimated from sample data either by a single number (a point estimate) or an entire interval of plausible values (a confidence interval). Frequently, however, the objective of an investigation is not to estimate a parameter but to decide which of two contradictory claims about the parameter is correct. Methods for accomplishing this comprise the part of statistical inference called hypothesis testing. In this chapter, we first discuss some of the basic concepts and terminology in hypothesis testing and then develop decision procedures for the most frequently encountered testing problems based on a sample from a single population.

Supplementary Exercises: (83–102)

• 83. When a drug is recalled for safety concerns (e.g., too many people having serious adverse reactions), the pharmaceutical company making the drug can only re-issue it by convincing the FDA that the reformulated version of the drug is safer than the original version.

  • a. In words, what are the null and alternative hypotheses for this situation? [Hint: the FDA will not allow re-issuance unless they see convincing evidence of a safety improvement.]

  • b. Describe the possible type I and type II errors in this scenario.

  • c. Which of the two possible errors is worse, and why? On that basis, how should the FDA determine the α level for testing whether the reformulated drug is safer?

• 84. A sample of 50 lenses used in eyeglasses yields a sample mean thickness of 3.05 mm and a sample standard deviation of .34 mm. The desired true average thickness of such lenses is 3.20 mm. Does the data strongly suggest that the true average thickness of such lenses is something other than what is desired? Test using α = .05.

• 85. In the previous exercise, suppose the experimenter had believed before collecting the data that the value of σ was approximately .30. If the experimenter wished the probability of a type II error to be .05 when μ = 3.00, was a sample size of 50 unnecessarily large?

• 86. It is specified that a certain type of iron should contain .85 g of silicon per 100 g of iron (.85%). The silicon content of each of 25 randomly selected iron specimens was determined, and the accompanying output resulted from a test of the appropriate hypotheses.

  Variable	N	Mean	St Dev	SE Mean	T	P
  sil cont	25	0.8880	0.1807	0.0361	1.05	0.30

  • a. What hypotheses were tested?

  • b. What conclusion would be reached for a significance level of .05, and why? Answer the same question for a significance level of .10.

• 87. A hot-tub manufacturer advertises that with its heating equipment, a temperature of 100 °F can be achieved in at most 15 min. A random sample of 32 tubs is selected, and the time necessary to achieve a 100 °F temperature is determined for each tub. The sample average time and sample standard deviation are 17.5 min and 2.2 min, respectively. Does this data cast doubt on the company’s claim? Compute the P-value and use it to reach a conclusion at level .05 (assume that the heating-time distribution is approximately normal).

• 88. The true average breaking strength of ceramic insulators of a certain type is supposed to be at least 10 psi. They will be used for a particular application unless sample data indicates conclusively that this specification has not been met. A test of hypotheses using α = .01 is to be based on a random sample of ten insulators. Assume that the breaking-strength distribution is normal with unknown standard deviation. [Note: Software is required for this exercise.]

  • a. If the true standard deviation is .80, how likely is it that insulators will be judged satisfactory when true average breaking strength is actually only 9.5? Only 9.0?

  • b. What sample size would be necessary to have a 75% chance of detecting that H₀ is false when true average breaking strength is 9.5 when the true standard deviation is .80?

• 89. The article “Caffeine Knowledge, Attitudes, and Consumption in Adult Women” (J. Nutrit. Ed. 1992: 179–184) reports the following summary data on daily caffeine consumption for a sample of adult women: n = 47, $\bar{x} = 215\; {\rm{mg}}$, s = 235 mg, and range = 5 − 1176.

  • a. Does it appear plausible that the population distribution of daily caffeine consumption is normal? Is it necessary to assume a normal population distribution to test hypotheses about the value of the population mean consumption? Explain your reasoning.

  • b. Suppose it had previously been believed that mean consumption was at most 200 mg. Does the given data contradict this prior belief? Test the appropriate hypotheses at significance level .10 and include a P-value in your analysis.

• 90. The incidence of a certain type of chromosome defect in the U.S. adult male population is believed to be 1 in 75. A random sample of 800 individuals in U.S. penal institutions reveals 16 who have such defects. Can it be concluded that the incidence rate of this defect among prisoners differs from the presumed rate for the entire adult male population?

  • a. State and test the relevant hypotheses using α = .05. What type of error might you have made in reaching a conclusion?

  • b. What P-value is associated with this test? Based on this P-value, could H₀ be rejected at significance level .20?

• 91. In an investigation of the toxin produced by a certain poisonous snake, a researcher prepared 26 different vials, each containing 1 g of the toxin, and then determined the amount of antitoxin needed to neutralize the toxin. The sample average amount of antitoxin necessary was found to be 1.89 mg, and the sample standard deviation was .42. Previous research had indicated that the true average neutralizing amount was 1.75 mg/g of toxin. Does the new data contradict the value suggested by prior research? Test the relevant hypotheses using the P-value approach. Does the validity of your analysis depend on any assumptions about the population distribution of neutralizing amount? Explain.

• 92. The sample average unrestrained compressive strength for 45 specimens of a particular type of brick was computed to be 3107 psi, and the sample standard deviation was 188. The distribution of unrestrained compressive strength may be somewhat skewed. Does the data strongly indicate that the true average unrestrained compressive strength is less than the design value of 3200? Test using α = .001.

• 93. To test the ability of auto mechanics to identify simple engine problems, an automobile with a single such problem was taken in turn to 72 different car repair facilities. Only 42 of the 72 mechanics who worked on the car correctly identified the problem. Does this strongly indicate that the true proportion of mechanics who could identify this problem is less than .75? Compute the P-value and reach a conclusion accordingly.

• 94. Chapter 8 presented a CI for the variance σ² of a normal population distribution. The key result there was that the rv $\chi^{2} = (n - 1)S^{2} /\sigma^{2}$ has a chi-squared distribution with n − 1 df. Consider the null hypothesis $H_{0} {:}\;\sigma^{2} = \sigma_{0}^{2}$ (equivalently, σ = σ₀). Then when H₀ is true, the test statistic $\chi^{2} = (n - 1)S^{2} /\sigma_{0}^{2}$ has a chi-squared distribution with n − 1 df. If the relevant alternative is $H_{\rm{a}}{:}\;\sigma^{2} \; > \;\sigma_{0}^{2}$, rejecting H₀ if $\chi^{2} \ge \chi_{\alpha ,n - 1}^{2}$ gives a test with significance level α. To ensure reasonably uniform characteristics for a particular application, it is desired that the true standard deviation of the softening point of a certain type of petroleum pitch be at most .50 °C. The softening points of ten different specimens were determined, yielding a sample standard deviation of .58 °C. Does this strongly contradict the uniformity specification? Test the appropriate hypotheses using α = .01.

• 95. Referring to the previous exercise, suppose an investigator wishes to test H₀: σ² = .04 versus H_a: σ² < .04 based on n = 21 observations. The computed value of 20s²/.04 is 8.58. Place bounds on the P-value and then reach a conclusion at level .01.

• 96. When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with E(S) ≈ σ and V(S) ≈ σ²/(2n). We already know that in this case, for any n, $\bar{X}$ is normal with $E(\bar{X}) = \mu$ and $V(\bar{X}) = \sigma^{2} /n$.

  • a. Assuming that the underlying distribution is normal, what is an approximately unbiased estimator of the 99th percentile θ = μ + 2.33σ?

  • b. As discussed in Section 6.4, when the X_i’s are normal $\bar{X}$ and S are independent rvs (one measures location whereas the other measures spread). Use this to compute $V(\hat{\theta })$ and $\sigma_{{\hat{\theta }}}$ for the estimator $\hat{\theta }$ of part (a). What is the estimated standard error $\hat{\sigma }_{{\hat{\theta }}}?$

  • c. Write a test statistic for testing H₀: θ = θ₀ that has approximately a standard normal distribution when H₀ is true. If soil pH is normally distributed in a certain region and 64 soil samples yield $\bar{x} = 6.33$, s = .16, does this provide strong evidence for concluding that at most 99% of all possible samples would have a pH of less than 6.75? Test using α = .01.

• 97. Let X₁, X₂, …, X_n be a random sample from an exponential distribution with parameter λ. Then it can be shown that 2λ$\sum$X_i has a chi-squared distribution with ν = 2n (by first showing that 2λX_i has a chi-squared distribution with ν = 2).

  • a. Use this fact to obtain a test statistic and rejection region that together specify a level α test for H₀: μ = μ₀ versus each of the three commonly encountered alternatives. [Hint: E(X_i) = μ = 1/λ, so μ = μ₀ is equivalent to λ = 1/μ₀.]

  • b. Suppose that ten identical components, each having exponentially distributed time until failure, are tested. The resulting failure times are

    95

    16

    11

    3

    42

    71

    225

    64

    87

    123

    Use the test procedure of part (a) to decide whether the data strongly suggests that the true average lifetime is less than the previously claimed value of 75.

• 98. Suppose the population distribution is normal with known σ. Let γ be such that 0 < γ < α. For testing H₀: μ = μ₀ versus H_a: μ ≠ μ₀, consider the test that rejects H₀ if either z ≥ z_γ or z ≤ −z_α–γ, where the test statistic is $Z = (\bar{X} - \mu_{0} )/(\sigma /\sqrt n ).$

  • a. Show that P(type I error) = α.

  • b. Derive an expression for β(μ′). [Hint: Express the test in the form “reject H₀ if either $\bar{x} \ge c_{1} \;{\rm{or}}\; \le c_{2}$.”]

  • c. Let Δ > 0. For what values of γ (relative to α) will β(μ₀ + Δ) < β(μ₀ − Δ)?

• 99. After a period of apprenticeship, an organization gives an exam that must be passed to be eligible for membership. Let p = P(randomly chosen apprentice passes). The organization wishes an exam that most but not all should be able to pass, so it decides that p = .90 is desirable. For a particular exam, the relevant hypotheses are H₀: p = .90 versus H_a: p ≠ .90. Suppose ten people take the exam, and let X = the number who pass.

  • a. Does the lower-tailed region {0, 1, …, 5} specify a level .01 test?

  • b. Show that even though H_a is two-sided, no two-tailed test is a level .01 test.

  • c. Sketch a graph of power as a function of p′ for this test. Is this desirable?

• 100. A service station has six gas pumps. When no vehicles are at the station, let p_i denote the probability that the next vehicle will select pump i (i = 1, 2, …, 6). Based on a sample of size n, we wish to test H₀: $p_{1} = \cdots = p_{6}$ versus the alternative H_a: p₁ = p₃ = p₅, p₂ = p₄ = p₆ (note that H_a is not a simple hypothesis). Let X be the number of customers in the sample that select an even-numbered pump.

  • a. Show that the likelihood ratio test rejects H₀ if either X ≥ c or X ≤ n − c. [Hint: When H_a is true, let θ denote the common value of p₂, p₄, and p₆.]

  • b. Let n = 10 and c = 9. Determine the power of the test both when H₀ is true and also when $p_{2} = p_{4} = p_{6} = 1/10,\;p_{1} = p_{3} = p_{5} = 7/30.$

• 101. Consider testing a pair of simple hypotheses H₀: θ = θ₀ versus H_a: θ = θ_a. Rather than prescribing the significance level and minimizing P(type II error), imagine trying to minimize the linear combination $a \cdot \alpha + b \cdot \beta$ for some specified constants a > 0 and b > 0. Show that $a \cdot \alpha + b \cdot \beta$ is minimized by using the rejection region

  $$ R^{*} = \left\{ {(x_{1} , \ldots ,x_{n} ){\mkern 1mu} {:}{\mkern 1mu} \frac{{f(x_{1} , \ldots ,x_{n} ;\theta_{\rm{a}} )}}{{f(x_{1} , \ldots ,x_{n} ;\theta_{0} )}} \ge \frac{a}{b}} \right\} $$

  [Hint: Imitate the first half of the proof of the Neyman-Pearson Lemma, but use $a \cdot \alpha + b \cdot \beta$ in place of kα + β].

• 102. Refer back to the scenario introduced in Example 9.23, where H₀: μ = 1 versus H_a: μ = 2 was tested based on a sample from a Poisson(μ) distribution. Suppose committing a type II error is considered 3 times as problematic as a type I error, and so the manufacturers wish to minimize α + 3β.

  • a. Determine the test procedure that minimizes α + 3β when n = 5. [Hint: Refer back to the previous exercise.]

  • b. For the test procedure in part (a), what are α, β, and the (minimized) value of α + 3β?

  • c. Repeat parts (a)–(b) for n = 10.