Verified Answer :
- We test the null hypothesis \(H_0: p = 0.3\) against \(H_1: p > 0.3\) with \(n=20\) and observation \(x=10\).
- The p-value is \(P(X \ge 10 \mid p=0.3) \approx 0.048\).
- Since the p-value is \(\le 0.05\), we reject \(H_0\).
- For the observation \(x=3\) in part 4, we simply record the result.
- The final conclusion for part 3 is to reject the null hypothesis: \(\boxed{\text{Reject } H_0}\).
Explanation
- Analyze the data and objective
To analyze the given data, we consider a random variable \(X\) that follows a binomial distribution with \(n=20\) trials and a success probability \(p\), denoted as \(X \sim B(20, p)\).
For Part 3, we conduct a hypothesis test to determine if the true probability of success is greater than the value suggested by the null hypothesis. The null hypothesis is \(H_0: p = 0.3\), and our alternative hypothesis is \(H_1: p > 0.3\). We are given an observation \(x=10\) and a significance level of \(\alpha = 0.05\).
For Part 4, we have an observation \(x=3\) for the same distribution \(X \sim B(20, p)\).
- Calculate the p-value for Part 3
To test the hypothesis \(H_0: p = 0.3\) against \(H_1: p > 0.3\), we calculate the p-value. The p-value is the probability of observing a result at least as extreme as our observed value of \(x=10\), assuming the null hypothesis is true. This is given by:
\[P(X \ge 10 \mid p = 0.3) = \sum_{k=10}^{20} \binom{20}{k} (0.3)^k (0.7)^{20-k}\]
This calculation represents the right-tail probability of the distribution under the null hypothesis. Using the properties of the cumulative distribution function, we can compute this as:
\[P(X \ge 10) = 1 - P(X \le 9) = 1 - \sum_{k=0}^{9} \binom{20}{k} (0.3)^k (0.7)^{20-k}\]
The calculation yields a p-value of approximately \(0.048\).
- Interpret the results for Part 3 and conclude
Now we compare the p-value we calculated to our significance level \(\alpha = 0.05\).
- The p-value is \(0.048\).
- The significance level \(\alpha = 0.05\).
Since \(0.048 \le 0.05\), our p-value is less than or equal to the significance level. Therefore, we reject the null hypothesis \(H_0: p = 0.3\) in favor of the alternative hypothesis \(H_1: p > 0.3\). We have statistically significant evidence at the \(5\%\) level to conclude that \(p > 0.3\).
- Observe data for Part 4
For Part 4, we have a random variable \(X \sim B(20, p)\) with a single observed value \(x=3\). Since no specific hypothesis test or further instruction was provided for this observation, we simply note that the observed success count is \(3\). This observation would be used if we were, for example, estimating the parameter \(p\) using the method of moments (\(\hat{p} = \frac{x}{n} = \frac{3}{20} = 0.15\)).
Examples
In fields like quality control, you might test if a machine's defect rate is higher than a standard threshold. If you sample 20 products and find 10 defects, you can calculate the probability of seeing that many defects by chance (the p-value) under your standard assumption (e.g., 30% defect rate). If that probability is very low, it signals that the process may be underperforming and needs adjustment, helping you make informed decisions based on data.