In a random sample of 80 ears of corn, farmer Carl finds that 11 of them have worms. Carl claims that less than 20% of all his corn has worms. Test this claim at the 0.05 significance level.
(a) What is the test statistic?
zp̂ =
(b) What is the P-value of the test statistic?
P-value =
(c) What is the critical value of z?
zα =
To test Carl's claim, we can use the hypothesis testing framework.
Step 1: State the hypotheses
- Null Hypothesis (H0): The proportion of corn with worms is greater than or equal to 20%.
- Alternative Hypothesis (Ha): The proportion of corn with worms is less than 20%.
Step 2: Select a significance level
The given significance level is 0.05, which means we want to be 95% confident in our conclusion.
Step 3: Collect the sample data
In this case, we are given that in a random sample of 80 ears of corn, 11 have worms.
Step 4: Compute the test statistic
The test statistic for this hypothesis test is the z-score. The formula for calculating the test statistic, zp̂, is:
zp̂ = (p̂ - p) / sqrt(p*(1-p)/n)
Where:
p̂ is the sample proportion of corn with worms, which is calculated as 11/80 = 0.1375
p is the hypothesized proportion of corn with worms under the null hypothesis, which is 0.20 (20%)
n is the sample size, which is 80.
Substituting the values into the formula:
zp̂ = (0.1375 - 0.20) / sqrt(0.20 * (1-0.20) / 80)
= -0.0625 / sqrt(0.16 / 80)
= -0.0625 / sqrt(0.002)
Hence, the test statistic (zp̂) is -0.0625 / 0.0447 = -1.3979 (rounded to four decimal places).
Step 5: Compute the P-value
The P-value is the probability of observing a test statistic as extreme as the one calculated (in either tail of the distribution) if the null hypothesis is true. In this case, we are performing a one-tailed test, since we are testing for the proportion of corn with worms being less than 20%.
To find the P-value, we need to find the area under the standard normal curve to the left of the test statistic zp̂. Using a standard normal distribution table or a statistical calculator, we find that the P-value is 0.0813 (rounded to four decimal places).
Step 6: Make a decision
Compare the P-value to the significance level.
- If the P-value is less than the significance level (0.05), we reject the null hypothesis in favor of the alternative hypothesis.
- If the P-value is greater than or equal to the significance level, we fail to reject the null hypothesis.
In this case, the P-value (0.0813) is greater than the significance level (0.05). Therefore, we fail to reject the null hypothesis.
Step 7: Interpret the results
Based on the data and hypothesis test, there is not enough evidence to support Carl's claim that less than 20% of all his corn has worms.