A biologist knows that the average length of a leaf of a certain full-grown plant is 4 inches. The standard deviation of the population is 0.6 inch. A sample of 20 leaves of that type of plant given a new type of plant food had an average length of 4.2 inches. Is there reason to believe that the new food is responsible for a change in the growth of the leaves? Use α = 0.01.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. How does it compare to .01?
To determine whether the new food is responsible for a change in the growth of the leaves, we can conduct a hypothesis test.
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The new plant food has no effect on the growth of the leaves.
- Alternative hypothesis (H1): The new plant food has an effect on the growth of the leaves.
Step 2: Determine the test statistic:
We will use a one-sample z-test to compare the sample mean to the population mean.
The formula for the z-test statistic is:
z = (x̄ - μ) / (σ / √n)
Where:
x̄ is the sample mean (4.2 inches)
μ is the population mean (4 inches)
σ is the population standard deviation (0.6 inches)
n is the sample size (20)
Step 3: Calculate the test statistic:
z = (4.2 - 4) / (0.6 / √20) = 1.732
Step 4: Determine the critical value:
The critical value is the value beyond which we reject the null hypothesis.
Since α (significance level) is given as 0.01, we need to find the z-value that corresponds to a cumulative probability of 0.99 (1 - α).
Using a z-table or statistical software, the critical value for α = 0.01 is approximately 2.326.
Step 5: Compare the test statistic to the critical value:
Since the test statistic (1.732) is less than the critical value (2.326), it does not fall into the rejection region.
Step 6: Make a decision:
Since the test statistic does not fall into the rejection region, we fail to reject the null hypothesis.
Step 7: Conclusion:
Based on the given data and significance level, there is not enough evidence to conclude that the new plant food has an effect on the growth of the leaves.
To determine whether there is reason to believe that the new food is responsible for a change in the growth of the leaves, we can use a hypothesis test.
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis: The new food has no effect on the growth of the leaves. The average length of the leaves is still 4 inches.
- Alternative hypothesis: The new food has an effect on the growth of the leaves. The average length of the leaves is different from 4 inches.
Step 2: Set the significance level (α):
- The significance level, denoted as α, is the maximum probability of making a Type I error (rejecting the null hypothesis when it is true). In this case, α = 0.01.
Step 3: Calculate the test statistic:
- The test statistic, known as the z-score, measures how many standard deviations the sample mean is away from the population mean.
- The formula for the z-score is: z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, x = 4.2, μ = 4, σ = 0.6, and n = 20.
Plugging in these values, we get: z = (4.2 - 4) / (0.6 / √20) = 1.7889.
Step 4: Determine the critical value:
- The critical value is the value that separates the rejection region (where we reject the null hypothesis) from the non-rejection region (where we fail to reject the null hypothesis).
- The critical value can be found using a z-table or a statistical calculator. Since we are testing for a two-tailed hypothesis, we divide the significance level (α = 0.01) by 2 to get α/2 = 0.005. Looking up this value in the z-table, we find that the critical value is approximately ±2.576.
Step 5: Compare the test statistic with the critical value:
- If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the absolute value of the test statistic (|1.7889| = 1.7889) is less than the critical value (2.576). Therefore, we fail to reject the null hypothesis.
Step 6: Interpret the result:
- Since we failed to reject the null hypothesis, there is not enough evidence to suggest that the new food is responsible for a change in the growth of the leaves. The average length of the leaves remains 4 inches.
In conclusion, based on the given data and using a significance level of α = 0.01, there is not enough evidence to believe that the new food is responsible for a change in the growth of the leaves.