For a certain yes/no survey question, 55% of the respondents answered “yes”.
The null hypothesis is that the population is equally divided on this issue (50% yes and 50% no).
The sample size is equal to your age.
For a significance level (alpha) of 5%, are you going to reject or accept the null hypothesis ?
What is the p-value ?
To determine whether you should reject or accept the null hypothesis, you need to perform a hypothesis test and calculate the p-value. Here's how you can do it:
1. Set up the null and alternative hypotheses:
- Null hypothesis (H0): The population is equally divided on this issue (50% yes and 50% no).
- Alternative hypothesis (Ha): The population is not equally divided (either more than 50% yes or less than 50% yes).
2. Calculate the test statistic:
- In this case, you can use the z-test statistic since you have a sample proportion and want to compare it to a specific value (50% yes).
- The formula for the z-test statistic is:
z = (p - P) / sqrt((P(1 - P)) / n)
- where p is the sample proportion, P is the population proportion under the null hypothesis, and n is the sample size.
3. Calculate the p-value:
- The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
- To find the p-value, you can use a z-table or a statistical software. The p-value represents the area under the curve of the standard normal distribution that is more extreme (greater than or equal to the observed z-value or less than or equal to the negative of the observed z-value).
4. Compare the p-value with the significance level (alpha):
- The significance level (alpha) represents the threshold below which you reject the null hypothesis.
- A common alpha level is 0.05, which corresponds to a 5% significance level.
- If the p-value is less than or equal to the significance level (p ≤ alpha), you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.
Since we don't know the sample size (your age), let's assume it is 100 for the sake of calculation.
Let's calculate the p-value using the formula mentioned earlier:
- p = 0.55 (given)
- P = 0.50 (specified under the null hypothesis)
- n = 100 (assumed sample size)
Using these values, we can calculate the z-test statistic:
z = (0.55 - 0.50) / sqrt((0.50 * (1 - 0.50)) / 100)
= 0.05 / sqrt(0.25 / 100)
= 0.05 / 0.05
= 1
The z-value we obtained is 1. Now, we need to find the area under the curve (p-value) corresponding to this z-value.
When we look up in the z-table, the two-tailed p-value for z = 1 is approximately 0.3172.
Finally, we compare the p-value (0.3172) with the significance level (0.05). Since the p-value is greater than the significance level (0.3172 > 0.05), we fail to reject the null hypothesis.
Therefore, for a sample size equal to your age, and a significance level of 5%, you would accept the null hypothesis, and the p-value is approximately 0.3172.