for a sample of 12 items from a normally distributed population for which the standard deviation is = 17.0, the sample mean is 230.8. At the 0.05 level of significance, test Ho : μ ≤ 220 versus Hı: μ >220. Determine and interpret the p-value for the test.
Using the z-test formula to find the test statistic:
z = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)
z = (230.8 - 220)/(17/√12)
Finish the calculation. Since this sample is from a normally distributed population, use a z-table to determine your p-value (the p-value is the actual level of the test statistic). Compare the p-value to 0.05 level of significance for a one-tailed test (the test is one-tailed because H1 is showing a specific direction). Determine whether or not to reject the null based on your outcome. If the null is rejected, then you conclude µ > 220. If the null is not rejected, then you cannot conclude a difference.
I hope this will help get you started.
To determine and interpret the p-value for the given hypothesis test, we need to follow these steps:
Step 1: State the hypotheses:
Ho: μ ≤ 220 (null hypothesis)
Hı: μ > 220 (alternative hypothesis)
Step 2: Set the significance level (alpha):
The significance level (α) is given as 0.05 or 5%.
Step 3: Compute the test statistic:
We will use the one-sample t-test because the population standard deviation is unknown. The formula for the t-test statistic is given by:
t = (x̄ - μ)/(s / sqrt(n))
where x̄ is the sample mean (230.8), μ is the hypothesized population mean (220), s is the sample standard deviation (17.0), and n is the sample size (12).
Plugging in the values:
t = (230.8 - 220) / (17 / sqrt(12))
= 10.8 / (17 / sqrt(12))
= 10.8 / (17 / 3.464)
= 10.8 / 4.969
= 2.17 (rounded to two decimal places)
Step 4: Determine the critical value:
Since the alternative hypothesis is μ > 220, we have a right-tailed test. At the 0.05 level of significance and 11 degrees of freedom (n-1), the critical value can be obtained from the t-distribution table. For this case, it is approximately 1.796.
Step 5: Calculate the p-value:
The p-value is the probability of observing a test statistic as extreme as the one calculated (2.17) or more extreme, given that the null hypothesis is true.
Using a t-distribution table or software, we can find the p-value associated with a t-value of 2.17 and 11 degrees of freedom. The p-value turns out to be approximately 0.0297.
Step 6: Make a decision:
Compare the p-value with the significance level (α).
If the p-value is less than α (0.05), we reject the null hypothesis.
If the p-value is greater than α (0.05), we fail to reject the null hypothesis.
In this case, the p-value (0.0297) is less than α (0.05), so we reject the null hypothesis.
Interpretation:
The p-value of approximately 0.0297 suggests that there is strong evidence to reject the null hypothesis (Ho: μ ≤ 220). This means that the sample mean (230.8) is significantly greater than the hypothesized population mean of 220 at the 0.05 level of significance.
To determine and interpret the p-value for the given hypothesis test, we can follow these steps:
Step 1: State the hypotheses:
- Null Hypothesis (Ho): The population mean (μ) is less than or equal to 220.
- Alternative Hypothesis (Hı): The population mean (μ) is greater than 220.
Step 2: Determine the test statistic:
Since we know the sample mean (x̄), the population standard deviation (σ), and the sample size (n), we can use the formula for the test statistic:
t = (x̄ - μ) / (σ / sqrt(n))
In this case:
x̄ = 230.8
μ = 220
σ = 17.0
n = 12
Substituting the values:
t = (230.8 - 220) / (17.0 / sqrt(12))
Calculate the value of t using a calculator or software, which comes out to be approximately 1.145.
Step 3: Determine the critical value:
Since the alternative hypothesis is μ > 220, this is a right-tailed test. Using the significance level, α = 0.05, and the degrees of freedom, df = n - 1 = 12 - 1 = 11, we can find the critical value from the t-distribution table or software. The critical value will be the value associated with the 0.95 percentile or the t-value with 11 degrees of freedom at the 0.05 level of significance.
Step 4: Calculate the p-value:
The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. Since this is a right-tailed test and the test statistic falls in the rejection region (t > critical value), the p-value is the probability of getting a t-value greater than the observed t-value.
Using a t-distribution calculator or software, calculate the probability of obtaining a t-value greater than 1.145, given the degrees of freedom (11). The p-value comes out to be approximately 0.141.
Step 5: Interpret the p-value:
The p-value represents the strength of evidence against the null hypothesis based on the observed data. In this case, the obtained p-value of 0.141 means that there is not sufficient evidence to reject the null hypothesis at the 0.05 level of significance. The p-value is greater than the significance level (α), indicating that we fail to reject the null hypothesis. Therefore, there is not enough evidence to suggest that the population mean is greater than 220 based on the given sample.
Please note that you may need to use statistical software or a t-distribution table to perform the calculations accurately.