Statistics: Many track hurdlers believe that starting in the inside lane closest to the field has better chance of winning. closest to the field is lane 1, nxt is lane 2, etc until Lane 6. find the critical value x20 to test the claim that the possibilities of winnin are the same in different positions. Use alpha = .05. Results are based on 240 wins. Starting position 1 2 3 4 5 6,Number of wins 32 45 50 36 33 44
To test the claim that the possibilities of winning are the same in different positions, we can use the chi-squared test for independence.
1. Step one: Formulate the null and alternative hypotheses.
- Null hypothesis (H0): The probabilities of winning are the same in different positions.
- Alternative hypothesis (Ha): The probabilities of winning are different in different positions.
2. Step two: Determine the level of significance (alpha).
In this case, alpha is given as 0.05.
3. Step three: Set up the decision rule.
Since the test statistic for the chi-squared test follows a chi-squared distribution, we need to find the critical value from the chi-squared table based on the degrees of freedom and the chosen significance level.
The degrees of freedom for this test are calculated as (n-1), where n is the number of categories or positions.
In our case, there are 6 positions, so the degrees of freedom (df) = 6 - 1 = 5.
Looking up the chi-squared critical value for df = 5 and alpha = 0.05, we find that the critical value (x20) is approximately 11.070.
4. Step four: Compute the test statistic (chi-squared statistic).
Next, we need to calculate the chi-squared statistic using the observed and expected frequencies.
Observed frequencies:
Position 1: 32
Position 2: 45
Position 3: 50
Position 4: 36
Position 5: 33
Position 6: 44
Expected frequencies:
To determine the expected frequencies, we assume that the probabilities of winning are the same for all positions.
The total number of wins is 240. To find the expected frequency for each position, we divide the total wins by the number of positions.
Expected frequency = Total wins / Number of positions
Expected frequency = 240 / 6 = 40
Position 1: 40
Position 2: 40
Position 3: 40
Position 4: 40
Position 5: 40
Position 6: 40
Now we can calculate the chi-squared statistic using the formula:
chi-squared = ∑ ( (Observed frequency - Expected frequency)² / Expected frequency )
chi-squared = ( (32-40)²/40 ) + ( (45-40)²/40 ) + ( (50-40)²/40 ) + ( (36-40)²/40 ) + ( (33-40)²/40 ) + ( (44-40)²/40 )
5. Step five: Make a decision.
Compare the computed chi-squared statistic with the critical value. If the computed chi-squared statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
If chi-squared > x20, then reject H0.
If chi-squared <= x20, then fail to reject H0.
If the null hypothesis is rejected, it means that the probabilities of winning are different in different positions.
I'm Sorry but it seems like the data you gave is incomplete as it only provides the number of wins in each position and not the total number of races or trials for each position. To perform the chi-squared test, we need the frequencies of both wins and losses for each position. Please provide the total number of races or trials for each position, and I'll be able to assist you further.
To test the claim that the possibilities of winning are the same in different positions, we can use a chi-square goodness-of-fit test. In this case, we will be comparing the observed number of wins in each position to the expected number of wins if the positions were equally likely.
First, let's calculate the expected number of wins in each position. Since there are 6 positions and a total of 240 wins, the expected number of wins in each position would be 240/6 = 40.
Now we can set up our hypotheses:
- Null hypothesis (H0): The possibilities of winning are the same in different positions.
- Alternate hypothesis (Ha): The possibilities of winning are not the same in different positions.
Next, we calculate the chi-square statistic using the following formula:
χ^2 = ∑ ((Observed - Expected)^2 / Expected)
Using the given data, we have:
Position: 1 2 3 4 5 6
Observed: 32 45 50 36 33 44
Expected: 40 40 40 40 40 40
χ^2 = [(32-40)^2/40] + [(45-40)^2/40] + [(50-40)^2/40] + [(36-40)^2/40] + [(33-40)^2/40] + [(44-40)^2/40]
Calculate this expression further to get the value of χ^2.
Now we need to find the critical value, which is denoted as χ^2_alpha/2, df (degrees of freedom). In this case, alpha is 0.05 and df is 6-1 = 5.
Look up the critical value in the chi-square distribution table for 5 degrees of freedom, with an alpha level of 0.05. The critical value represents the point at which we reject the null hypothesis.
Compare the calculated chi-square value to the critical value. If the calculated chi-square value is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Please use this information and the formula provided to calculate the chi-square value and find the critical value from the chi-square distribution table.