A researcher claims that at least 10% of all football helmets have manufacturing flaws that could potentially cause injury to the wearer. A sample of 200 helmets revealed that 24 helmets contained such defects.
a. Does this finding support the researcher’s claim? Used a fixed-level test with α = 0.01
b. Find the P-value for this test.
To determine if the researcher's claim is supported, we can conduct a hypothesis test.
a. Hypotheses:
- Null hypothesis (H0): The proportion of helmets with manufacturing flaws is less than or equal to 10% (p ≤ 0.10).
- Alternative hypothesis (Ha): The proportion of helmets with manufacturing flaws is greater than 10% (p > 0.10).
We will use a one-tailed test at a significance level (α) of 0.01.
b. Calculation of the test statistic and critical value:
Based on the sample, we can calculate the sample proportion of helmets with defects:
p̂ = 24/200 = 0.12 (proportion of defects in the sample).
The test statistic for a proportion is given by:
Z = (p̂ - p) / √(p(1-p)/n)
Where:
p = 0.10 (hypothesized proportion of defects)
n = 200 (sample size)
Substituting the values:
Z = (0.12 - 0.10) / √(0.10(1-0.10)/200)
Z ≈ 0.02 / √(0.09/200)
Z ≈ 0.02 / 0.0212
Z ≈ 0.943
To determine the critical value, we need to find the z-score corresponding to an α of 0.01 and a one-tailed test. Looking up the z-score in a standard normal distribution table or using statistical software, we find:
Critical value (Zα) ≈ 2.33
c. Decision:
Since our test statistic (0.943) is not greater than the critical value (2.33), we fail to reject the null hypothesis. This means there is not enough evidence to support the claim that at least 10% of all football helmets have manufacturing flaws that could potentially cause injury to the wearer.
b. Calculation of the P-value:
The P-value is the probability of obtaining a test statistic as extreme as the one observed (or more extreme) if the null hypothesis is true.
The P-value for this test can be found by calculating the area to the right of the observed test statistic (0.943) in the standard normal distribution.
P-value ≈ 1 - Area to the left of 0.943 ≈ 1 - 0.827 ≈ 0.173
Therefore, the P-value for this test is approximately 0.173.
To determine whether or not the finding supports the researcher's claim, we need to perform a hypothesis test.
a. Null Hypothesis (H0): The proportion of football helmets with manufacturing flaws is less than or equal to 10% (p ≤ 0.10).
Alternative Hypothesis (Ha): The proportion of football helmets with manufacturing flaws is greater than 10% (p > 0.10).
We can use a one-sample proportion test to test the hypotheses.
To do this, we need to calculate the test statistic (z-score) using the formula:
z = (p̂ - p) / sqrt((p * (1-p))/n)
Where:
p̂ is the sample proportion of helmets with manufacturing flaws (24/200 = 0.12),
p is the proportion stated in the null hypothesis (0.10),
n is the sample size (200).
Calculating the z-score:
z = (0.12 - 0.10) / sqrt((0.10 * (1-0.10))/200)
Next, we need to find the critical value for the fixed-level test with α = 0.01. Since this is a one-tailed test (the alternative hypothesis is one-sided), we use the z-table to find the critical z-value.
Looking up the z-value for α = 0.01 in the z-table, we find that it is approximately 2.33 (corresponding to a tail probability of 0.01).
Now, we can compare the calculated z-score to the critical z-value:
If the calculated z-score is greater than the critical z-value (2.33), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
b. To find the p-value for this test, we need to calculate the probability of getting the observed sample proportion (or a more extreme value) assuming the null hypothesis is true.
We can find the p-value by calculating the area to the right of the calculated z-score under the standard normal distribution curve using the z-table.
The p-value represents the probability of obtaining a result as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true.
To summarize:
a. To determine if the finding supports the researcher's claim, calculate the z-score and compare it to the critical z-value. If the calculated z-score is greater than the critical z-value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
b. To find the p-value, calculate the area to the right of the calculated z-score using the z-table.
You can use a one-sample proportional z-test for your data. (Test sample proportion = 24/200 and sample size = 200) Convert all fractions to decimals. Find the critical value in the appropriate table at .01 level of significance for a two-tailed test. Compare the test statistic you calculate to the critical value from the table. If the test statistic exceeds the critical value, reject the null. If the test statistic does not exceed the critical value, do not reject the null. You can draw your conclusions from there.
Note: The p-value is the actual level of the test statistic you calculate. Find using the appropriate table.