Hypothesis Testing Examples with Solutions

One-sample t-test on Velocity Observation Data

An exercise science student measures the velocity of tosses thrown by 10 cricket bowlers. The observations are below.

142.3 � �138.6 � �140.3 � �143.1 � �138.5
135.4 � �137.9 � �146.9 � �138.1 � �144.9

Suppose we now want to test if the mean velocity is greater than 80.

� � Carry out the test (either by hand or using software). In doing so, complete these steps:
� � Write the null and alternative hypotheses (note this is a one-sided test).

� � Write the rejection region, if we use ? = .05.

� � Calculate the test statistic, t0.

� � State your decision - either say that you reject the null hypothesis, or that you fail to reject the null hypothesis. Interpret the result in the context of the problem.

�

� � State how the p-value can be used to reach the decision of the hypothesis test in part (a).Calculate the test statistic of the Wilcoxon rank sum test. �You do not have to compare this to a rejection region, and you do not have to interpret the result.� � Waist-to-hip ratio was measured on 8 men just before they entered a weight loss program (time 1) and again 6 months after the program (time 2). The results are below:

Subject � � 1 � � � �2 � � � �3 � � � 4 � � � �5 � � � �6 � � � �7 � � � 8 ��
Time 1 � � � � 1.02 � � 1.01 � � �1.18 � � 0.82 � � 1.02 � � 1.05 � � 1.06 � � 0.83
Time 2 � � � � 1.00 � � 1.02 � � �1.13 � � 0.78 � � 1.06 � � 1.00 � � 1.08 � � 0.76

� � Calculate the difference for each subject (as time 1 value minus time 2 value), then calculate the mean and standard deviation of these differences.Using the information from part (a), calculate a 95% confidence interval for the difference of the means for the two time periods (i.e., for the mean difference).�

� � We want to use a paired t-test, at ? = .05, to test if the mean waist-to-hip ratio is smaller at time 2 than at time 1 (i.e., test if the program is effective). Carry out the test by hand or using software. A colleague looks at the printout and decides that we have proven that the program is not effective. Do you agree with his conclusion? Why or why not?

?
� � The number of heart beats in a 5-minute period are recorded for 25 women not using a certain drug, and 16 women using this drug.

No drug: � �293 � 297 � 300 � 302 � 306 � 311 � 312 � 314 � 318 ��
� � � � � � 320 � 322 � 323 � 324 � 324 � 326 � 327 � 331 � 333 ��
� � � � � � 335 � 339 � 342 � 345 � 345 � 351 � 356

� � Drug: � �330 � 331 � 335 � 337 � 339 � 343 � 346 � 349
� � � � 350 � 358 � 364 � 369 � 375 � 387 � 398 � 403

We want to test, at ? = .01, if there is a difference in the mean number of beats for the two group Write the null and alternative hypotheses.Analyze the data. First assume that the variances are equal, and then assume that they are not equal.State the test statistic and p-value of the tests and interpret the results.

�

� � Examine which of the two tests is more appropriate.

?
Use the REACTIONTIME data set (provided as a comma separated value file, a tab-delimited file, and an excel file). We want to know whether the mean reaction time is faster for males than for females.Using ? = .1, be sure to make the necessary adjustment for this one-sided test.�

� � Calculate the pooled estimate of variance, s_p^2. The vocabulary size is measured for eleven 4-year-old children. The results are below. We want to test, at ?= .05, whether the population median differs from 450.
� ��
149 � �279 � �339 � �388 � �418 � �421 � �441 � �460 � �474 � �487 � �492

� � Write the null and alternative hypotheses. Perform a sign test. Calculate the test statistic, find the p-value (using the Binomial distribution), and interpret the result of the test.

�

� � Calculate the test statistic for the Wilcoxon signed rank test and interpret the result.

calculate the test statistic for the Wilcoxon signed rank test, you can follow these steps:

1. Calculate the difference between each pair of observations and discard any pairs where the difference is zero.

2. Rank the absolute values of the differences, with the smallest absolute difference receiving a rank of 1, the next smallest receiving a rank of 2, and so on. If there are ties, assign the average rank to all tied observations.

3. Calculate the sum of the ranks for positive differences and the sum of the ranks for negative differences. Let's call these sums T+ and T-, respectively.

4. The test statistic for the Wilcoxon signed rank test is the smaller of T+ and T-. If T+ is smaller, the null hypothesis is rejected in favor of the alternative hypothesis that the median of the differences is greater than zero. If T- is smaller, the null hypothesis is rejected in favor of the alternative hypothesis that the median of the differences is less than zero. If T+ and T- are equal, the null hypothesis is not rejected.