Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In one of the races, the runners wear one brand of shoe and in the other a different brand. The brand of shoe they wear in which race is determined at random. All runners are timed and are asked to run their best in each race. The results (in minutes) are given below: Runner Brand 1 Brand 2 1 31.23 32.02 2 29.33 28.98 3 30.50 30.63 4 32.20 32.67 5 33.08 32.95 6 31.52 31.53 7 30.68 30.83 8 31.05 31.10 9 33.00 33.12 10 29.67 29.50 11 30.55 30.57 12 32.12 32.20 Use the sign test for matched pairs to determine if there is evidence that times using Brand 1 tend to be faster than times using Brand 2. (a) What are the hypotheses we wish to test? i. H0 : μ = 0 versus Ha : μ > 0, where μ =the mean of the differences in running times (Brand 2-Brand 1) for all runners who run this race twice wearing the two brands of shoes. ii. H0 : p = 1 2 versus Ha : p , 1 2 , where p = the proportion of running times using Brand 1 that are faster than times using Brand 2. iii. H0 : p = 1 2 versus Ha : p > 1 2 , where p = the proportion of running times using Brand 1 that are faster than times using Brand 2. iv. H0: population median =0 versus Ha: population median , 0, where the median of the differences in running times for all runners who run this race twice wearing the two brands of shoes is measured for Brand 2-Brand 1. (b) What is the (approximate) value of the P-value? (c) Determine which of the following statements is true. i. We would not reject the null hypothesis of no difference at the 0.10 level. ii. We would reject the null hypothesis of no difference at the 0.10 level but not at the 0.05 level. iii. We would reject the null hypothesis of no differe
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In
one of the races, the runners wear one brand of shoe and in the other a different brand. The
brand of shoe they wear in which race is determined at random. All runners are timed and
are asked to run their best in each race. The results (in minutes) are given below:
Runner Brand 1 Brand 2
1 31.23 32.02
2 29.33 28.98
3 30.50 30.63
4 32.20 32.67
5 33.08 32.95
6 31.52 31.53
7 30.68 30.83
8 31.05 31.10
9 33.00 33.12
10 29.67 29.50
11 30.55 30.57
12 32.12 32.20
Use the sign test for matched pairs to determine if there is evidence that times using Brand
1 tend to be faster than times using Brand 2.
(a) What are the hypotheses we wish to test?
i. H0 : μ = 0 versus Ha : μ > 0, where μ =the mean of the differences in running
times (Brand 2-Brand 1) for all runners who run this race twice wearing the two
brands of shoes.
ii. H0 : p = 1
2 versus Ha : p , 1
2 , where p = the proportion of running times using
Brand 1 that are faster than times using Brand 2.
iii. H0 : p = 1
2 versus Ha : p >
1
2 , where p = the proportion of running times using
Brand 1 that are faster than times using Brand 2.
iv. H0: population
of the differences in running times for all runners who run this race twice wearing
the two brands of shoes is measured for Brand 2-Brand 1.
(b) What is the (approximate) value of the P-value?
(c) Determine which of the following statements is true.
i. We would not reject the null hypothesis of no difference at the 0.10 level.
ii. We would reject the null hypothesis of no difference at the 0.10 level but not at the
0.05 level.
iii. We would reject the null hypothesis of no difference at the 0.05 level but not at the
0.01 level.
iv. We would reject the null hypothesis of no difference at the 0.01 level.
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