The international olympic committe states that the female participation in the 2004 summer olympic games was 42%, even with new sports such as weight lifting, hammer throw, and modern pentathalon being added to the games. Broadcasting and clothing companies want to change their advertising and marketing strategies if the female participation increases at the next games. An independent sports expert reported that 202 of 454 athletes in the random sample were women. is this strong evidence that the participation rate may increase?
1) test an appropriate hypothesis and state your conclusion.
2) clearly explain what a Type-I error is in this context and give a possible consequence of making a type-I error.
3) clearly explain what a type II error is in this context and give a possible consequence of making a type-II error.
4) clealry explain, in context, what your P-value means
You can try a proportional one-sample z-test for this one since this problem is using proportions.
Here's a few hints to get you started:
Null hypothesis:
Ho: p = .42 -->meaning: population proportion is equal to .42
Alternative hypothesis:
Ha: p > .42 -->meaning: population proportion is greater than .42
Using a formula for a proportional one-sample z-test with your data included, we have:
z = .44 - .42 -->test value (202/454 is approximately .44) minus population value (.42) divided by
√[(.42)(.58)/454] --> .58 represents 1-.42 and 454 is sample size.
Finish the calculation. Remember if the null is not rejected, then there is no difference. If you need to find the p-value for the test statistic, check a z-table. The p-value is the actual level of the test statistic.
I'll give you some background on Type I and Type II errors and let you take it from there.
Type I errors result when you reject the null and it's true. Type II errors result when you accept the null and it's false. You can reduce Type I errors by setting the alpha at a lower level, for example, from .05 to .01. However, when you do that, you increase the probability of making a Type II error. You would have to determine if the interested parties would be more concerned about Type I or Type II errors.
I hope these hints will help.
1) To test whether the participation rate may increase, we can set up a hypothesis test.
Null Hypothesis (H0): The participation rate remains the same or decreases.
Alternative Hypothesis (Ha): The participation rate increases.
Next, we need to calculate the test statistic and compare it to the critical value or p-value to make a decision.
Let's assume a significance level of alpha = 0.05. Using the reported data, we can calculate the test statistic and p-value.
The observed proportion of female athletes in the sample is 202/454 = 0.445 (rounded to three decimal places).
Under the null hypothesis, the expected proportion of female athletes is 42% or 0.42.
We can now use a one-sample proportion test (Z-test) to compare the observed and expected proportions.
The formula for the test statistic (Z-score) is:
Z = (p - P0) / sqrt [(P0 * (1 - P0)) / n]
Where:
p = observed proportion (0.445)
P0 = expected proportion under H0 (0.42)
n = sample size (454)
Calculating the test statistic:
Z = (0.445 - 0.42) / sqrt [(0.42 * (1 - 0.42)) / 454]
After performing the calculations, we find the test statistic Z is approximately 1.325 (rounded to three decimal places).
From the Z-table or using statistical software, we find that the critical Z-value for an alpha of 0.05 (two-tailed test) is approximately ±1.96.
Since the calculated Z-value (1.325) is less than the critical Z-value (1.96), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that the participation rate in the next games may increase based on this sample.
2) Type-I error, in this context, refers to rejecting the null hypothesis (H0) when it is actually true. In other words, it's a false positive.
A possible consequence of making a Type-I error would be that broadcasting and clothing companies change their advertising and marketing strategies assuming an increase in female participation, leading to higher costs and potentially missed opportunities to target the correct audience if the participation rate does not actually increase.
3) Type II error, in this context, refers to accepting the null hypothesis (H0) when it is actually false. It is a false negative.
A possible consequence of making a Type II error would be that broadcasting and clothing companies do not change their advertising and marketing strategies assuming the participation rate will not increase. If the participation rate does indeed increase, they could miss opportunities to target the growing female audience effectively and generate more sales.
4) In hypothesis testing, the P-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
A small P-value indicates that the observed data is unlikely under the null hypothesis and suggests strong evidence against the null hypothesis. Conversely, a larger P-value suggests that the observed data is reasonably likely to occur under the null hypothesis.
In this context, if we obtain a low P-value (e.g., less than the chosen significance level), it would indicate strong evidence that the participation rate may increase. On the other hand, if we obtain a high P-value, it suggests weak evidence against the null hypothesis, and we would fail to reject the null hypothesis, indicating the participation rate may not increase.