A medical research team studied the number of head and neck injuries sustained by hockey players. Of the 200 players who wore a full-faced shield, 40 sustained an injury. Of the 250 players who wore a half-faced shield, 60 sustained an injury. At á = 0.05, can you reject the claim that the proportion of the players sustaining head and neck injuries are the same for the two groups?
Use a formula for a binomial proportion two-sample z-test for this one since you have two groups.
Formula:
z = (p1 - p2)/√[pq(1/n1 + 1/n2)]
...note: 'n' represents the sample sizes, 'p' is (x1 + x2)/(n1 + n2), and 'q' is 1-p.
I'll get you started:
p = (40 + 60)/(200 + 250) = ? -->once you have the fraction, convert to a decimal (decimals are easier to use in the formula)
p1 = 40/200
p2 = 60/250
Convert all fractions to decimals. Plug those decimal values into the formula and find z. Once you have this value, compare to the critical or cutoff value you find in a z-table at 0.05 level of significance for a two-tailed test. If the test statistic exceeds the critical value in the table, reject the null and conclude a difference between the two groups. If the test statistic does not exceed the critical value from the table, you cannot reject the null and you cannot conclude a difference.
I hope this will help get you started.
To determine whether you can reject the claim that the proportion of players sustaining head and neck injuries is the same for the two groups, you can perform a hypothesis test.
1. State the null and alternative hypotheses:
- Null hypothesis (H₀): The proportion of players sustaining head and neck injuries is the same for the two groups.
- Alternative hypothesis (H₁): The proportion of players sustaining head and neck injuries differs for the two groups.
2. Define the significance level (α):
- In this case, α (also denoted as á) is given as 0.05. This means that we are willing to accept a 5% chance of mistakenly rejecting the null hypothesis when it is actually true.
3. Calculate the test statistic and the critical value:
- We will use the two-sample Z-test to compare the proportions.
- First, calculate the proportion of injuries for each group:
- For the group wearing full-faced shields: p₁ = 40/200 = 0.2
- For the group wearing half-faced shields: p₂ = 60/250 = 0.24
- The test statistic is computed using the formula:
- Z = (p₁ - p₂) / √[(p̂ * q̂ * (1/n₁ + 1/n₂))]
where p̂ = (x₁ + x₂) / (n₁ + n₂), q̂ = 1 - p̂, x₁ and x₂ are the respective injury counts, and n₁ and n₂ are the respective sample sizes.
- The critical value for a two-tailed test with α = 0.05 is ±1.96.
4. Evaluate the test statistic:
- Calculate the test statistic using the values:
- Z = (0.2 - 0.24) / √[(0.2 * 0.8 * (1/200 + 1/250))]
≈ -1.21
- The calculated test statistic is -1.21.
5. Compare the test statistic with the critical value:
- The critical value for a two-tailed test with α = 0.05 is ±1.96.
- Since -1.96 < -1.21 < 1.96, the test statistic falls within the non-rejection region.
6. Make a decision:
- Because the test statistic falls within the non-rejection region, you fail to reject the null hypothesis.
- In other words, at a significance level of 0.05, there is not enough evidence to claim that the proportion of players sustaining head and neck injuries is different for the two groups.
Therefore, you cannot reject the claim that the proportion of players sustaining head and neck injuries is the same for the two groups at á = 0.05.