H0: pi<=.60 versus H1:pi>=.60, alpha:.05
x:56
n:80
I really don't get where Alpha is involved, pls help.
Thanks
In hypothesis testing, the alpha (α) level is the predetermined level of significance that determines the threshold for rejecting the null hypothesis (H0). It is the probability of making a Type I error, which is the incorrect rejection of a true null hypothesis.
In your example, the alpha level is given as 0.05, which means you have set a strict criterion for rejecting the null hypothesis. If the p-value (the probability of obtaining a test statistic as extreme as observed or more extreme, assuming the null hypothesis is true) is less than or equal to 0.05, you will reject the null hypothesis in favor of the alternative hypothesis (H1).
Now, let's apply this to your specific hypothesis test:
H0: π ≤ 0.60 (Null hypothesis)
H1: π > 0.60 (Alternative hypothesis)
In this case, you want to test whether the proportion (π) is less than or equal to 0.60 or greater than 0.60. The observed sample proportion (x/n) is given as 56/80.
To perform the hypothesis test, you need to calculate the test statistic and compare it to the critical value or p-value to make a decision.
The test statistic (Z) is calculated using the formula:
Z = (p̂ - π0) / √(π0 * (1 - π0) / n)
Where:
p̂ is the observed sample proportion (x/n)
π0 is the value under the null hypothesis (H0)
n is the sample size
In your case, π0 is 0.60, p̂ is 56/80, and n is 80. Therefore, the test statistic can be calculated as follows:
Z = ((56/80) - 0.60) / √(0.60 * (1 - 0.60) / 80)
Once you have the test statistic, you can find the corresponding p-value using a standard normal distribution table or statistical software.
If the p-value is less than or equal to 0.05 (the alpha level), you reject the null hypothesis. If the p-value is greater than 0.05, you fail to reject the null hypothesis.
I hope this explanation helps you understand how the alpha level is involved in hypothesis testing.