Reading 3

In a hypothesis test, suppose there is not sufficient evidence to reject the null hypothesis. Which statements are true?

The null hypothesis might be true. The evidence was not strong enough to reject the null hypothesis.

The P-value of a hypothesis test is the probability, under H0, of ______.

a test statistic as extreme or more extreme than the one observed

A hypothesis test result can have statistical significance without having Blank significance.

practical

At the beginning of a hypothesis test, which hypothesis is assumed to be true?

Ho

A random sample of 4 weighings of a gold nugget using a cold balance had xx = 21.4 grams, with sX = 0.013 grams. A random sample of 6 more weighings of the same nugget using the balance after it warmed up had yy = 21.3, with sY = 0.012. Supposing that repeated weighings are normally distributed, and that σX = σY without regard for the balance's temperature, the test statistic for H0: μcold - μwarm = 0 is ______.

(21.4-21.3)−0 / [(4-1)(0.0132)+(6-1)(0.0122) / 4+6-2]√1/4+1/6√

Independent random samples of forged nails were drawn from those produced by a blacksmith over two days. The first sample of 40 nails had mean length 60.2 mm, with standard deviation 1.1 mm. The second sample of 38 nails had mean 60.1 with standard deviation 0.9. For a test of the hypothesis that the two population means are equal, the test statistic is ______.

(60.2-60.1)-0 / 1.12/40+0.92/38√

To test H0: μX - μY = 0 using small independent samples requires which assumptions?

Each population is normal. The samples are random.

True or false: If the P-value is very small, we can be sure that the results have practical significance.

False

Suppose a two-sided level 100%(1 - α) confidence interval for μ contains μ0. Which statement is true?

For a test of H0: μ = μ0 versus H1: μ ≠ μ0, H0 is retained at level 100%α.

Select three equivalent statements from among the following statements about hypothesis tests.

H0 is rejected at level 100%α. The P-value is smaller than the significance level, 100%α. The test result is statistically significant.

Match the P-value on the left with the appropriate conclusion on the right.

P-value < 0.05 matches H0 is probably false, so H1 is probably true. P-value > 0.05 matches H0 is plausible. P-value < 10-8 matches H0 is almost certainly false, so H1 is almost certainly true. P-value > 10-8 matches There is not enough information to draw a conclusion. There is not enough information to draw a

If the data do not provide strong evidence against the null hypothesis, which of the following is true?

We do not reject the null hypothesis. The null hypothesis may or may not be true.

If the data provide strong evidence against the null hypothesis, we reject the Blank hypothesis and conclude that the Blank hypothesis is true.

null alternative

In a hypothesis test, which two numbers are compared when deciding whether to reject or retain H0?

the threshold significance level, 100%α the P-value

Which statements about the P-value of a hypothesis test are true?

If the the P-value is smaller than α, then H0 is rejected. The smaller the P-value, the stronger the evidence against H0. The P-value is the probability, under H0, of a test statistic as extreme or more extreme than the one observed.

To test H0: μ = μ0 versus H1: μ ≠ μ0 from a random sample X1, ... , Xn requires that at least one of two assumptions be made. Which two?

The sample size n is large. The population is normal.

In a small-sample test for the difference between two means, suppose that the two sample standard deviations are nearly equal. Which statement is true?

The test for which the population standard deviations are not known to be equal usually performs better.

To test H0: μ = μ0 versus H1: μ ≠ μ0 from a random sample X1, ... , Xn, a z test or a t test may be used. Which are differences between the two tests?

The test statistics are computed differently. The z test requires a known standard deviation σ, while the t test does not require σ. P-values are computed using different distributions.

A random sample of 5 Phillips Hall students had average weight 156 pounds, with standard deviation 17.3 pounds. A random sample of 4 Tripp Hall students had average weight 153, with standard deviation 14.2. Supposing that each hall's population of weights is normal, the test statistic for testing H0: μPhillips - μTripp = 0 is ______.

(156-153)-0 / 17.32/5+14.22/4√

A rule of thumb suggests rejecting H0 when the P-value ≤ ______.

0.05

In a large-sample test of the hypothesis that two population means are equal against the alternative that they are not equal, the test statistic was found to be z = 2.50. The P-value is computed as ______.

2P(Z ≥ 2.50)

Match the statement about a confidence interval on the left with the equivalent statement about a test on the right. (Here θ is a placeholder for a parameter like μ, σ, or p; and v is a constant.)

A level 100(1 - α)% confidence interval for θ contains v. matches H0: θ = v is not rejected at level 100α%. A level 100(1 - α)% confidence interval for θ excludes v. matches H0: θ = v is rejected at level 100α%. If a level 100(1 - α)% confidence interval for θ is found for each of many samples, 100α% of those intervals should fail to include θ. matches If H0:θ=v is true and a level 100α% test of H0 is done for each of many samples, H0 will be rejected 100α% of the time.

A random sample of beverage can volumes is measured to check the population mean volume, μ. The P-value is found to be 0.06 for a test of H0: μ = 355 ml versus H1: μ ≠ 355. Which statement is true?

A two-sided level 95% confidence interval for μ contains 355.

Suppose a ski run will open if we conclude that its average snow depth μ is greater than 0.5 m. The depth will be measured in 40 randomly chosen locations. Which pair of hypotheses is useful for this decision?

H0: μ ≤ 0.5 vs. H1: μ > 0.5

A cabin owner calls a pile of drying firewood ready to burn when its average water content is less than 20%. He measures the water content of a random sample of the logs in one of his piles. Which pair of hypotheses best helps him decide whether the pile is ready to burn?

H0: μ ≥ 0.2 vs. H1: μ < 0.2

Match the null hypothesis on the left to the corresponding alternate hypothesis on the right.

H0: μ ≥ μ0 matches H1: μ < μ0 H0: μ ≤ μ0 matches H1:μ>μ0 H0: μ = μ0 matches H1:μ≠μ0

Match the alternate hypothesis on the left to the corresponding P-value computation on the right.

H1: μ > μ0 matches P(Z ≥ z) H1: μ < μ0 matches P(Z≤z) H1: μ ≠ μ0 matches P(|Z|≥ |z|)

In a hypothesis test, suppose there is sufficient evidence to reject the null hypothesis. Which is the best conclusion?

The alternate hypothesis is true.

A graduate student performed a hypothesis test on data gathered by her thesis professor and decided to reject H0. In presenting the results to her professor, what information should she include?

The decision to reject H0 and the P-value

A manufacturer of 8 mm machine screws knows that the standard deviation of screw lengths is very close to σ = 0.06 mm, even when the mean length drifts away from the specified μ = 8 mm. The lengths are known to be approximately normally distributed. A random sample of 7 screws is measured. Which test statistic is best for testing H0: μ = 8 versus H1: μ ≠ 8?

Z = X-8 / 0.06/√7

A robot fills ice-cream cones whose mean weight tends to drift slowly from the specified μ = 147 g. Experience indicates the standard deviation of weights is steady around σ = 8 g and weights in any given day are normally distributed. The weights of a random sample of 11 cones from a day are found to have mean xx = 140 g and standard deviation s = 8.2 g. Which test statistic is best for testing H0: μ = 147 versus H1: μ ≠ 147?

z = (140-147) / (8/√11)