In Order For A Candy Company To Claim That A Bridge Mix Is Mostly Chocolate Stars, At Least 80% Of The Packages Must Contain 3 Ounces Or More Of Chocolate Stars. Quality Control Tests A Random Sample Of 50 Packages To Determine If The Proportion Is
In Order for a Candy Company to Claim a Bridge Mix is Mostly Chocolate Stars: A Statistical Analysis
In the world of food manufacturing, quality control is a crucial aspect of ensuring that products meet the required standards. For a candy company to claim that a bridge mix is mostly chocolate stars, it must adhere to specific regulations. One such regulation is that at least 80% of the packages must contain 3 ounces or more of chocolate stars. In this article, we will discuss how quality control tests a random sample of 50 packages to determine if the proportion of chocolate stars in the bridge mix meets the required standard.
The candy company wants to determine if the proportion of chocolate stars in the bridge mix is at least 80%. To do this, they conduct a quality control test on a random sample of 50 packages. Each package is weighed to determine the amount of chocolate stars it contains. The company then calculates the proportion of packages that contain 3 ounces or more of chocolate stars.
Let's assume that the quality control test yields the following data:
| Package | Weight of Chocolate Stars (oz) |
|---|---|
| 1 | 2.5 |
| 2 | 3.2 |
| 3 | 2.8 |
| 4 | 3.5 |
| 5 | 2.2 |
| 6 | 3.8 |
| 7 | 2.9 |
| 8 | 3.1 |
| 9 | 2.6 |
| 10 | 3.4 |
| 11 | 2.7 |
| 12 | 3.6 |
| 13 | 2.3 |
| 14 | 3.9 |
| 15 | 2.4 |
| 16 | 3.7 |
| 17 | 2.1 |
| 18 | 3.3 |
| 19 | 2.5 |
| 20 | 3.2 |
| 21 | 2.8 |
| 22 | 3.5 |
| 23 | 2.2 |
| 24 | 3.8 |
| 25 | 2.9 |
| 26 | 3.1 |
| 27 | 2.6 |
| 28 | 3.4 |
| 29 | 2.7 |
| 30 | 3.6 |
| 31 | 2.3 |
| 32 | 3.9 |
| 33 | 2.4 |
| 34 | 3.7 |
| 35 | 2.1 |
| 36 | 3.3 |
| 37 | 2.5 |
| 38 | 3.2 |
| 39 | 2.8 |
| 40 | 3.5 |
| 41 | 2.2 |
| 42 | 3.8 |
| 43 | 2.9 |
| 44 | 3.1 |
| 45 | 2.6 |
| 46 | 3.4 |
| 47 | 2.7 |
| 48 | 3.6 |
| 49 | 2.3 |
| 50 | 3.9 |
The candy company wants to determine if the proportion of packages that contain 3 ounces or more of chocolate stars is at least 80%. This can be expressed as a hypothesis:
H0: p ≤ 0.8 (The proportion of packages that contain 3 ounces or more of chocolate stars is less than or equal to 80%)
H1: p > 0.8 (The proportion of packages that contain 3 ounces or more of chocolate stars is greater than 80%)
To test the hypothesis, we need to calculate the test statistic. The test statistic is calculated as follows:
x̄ = (number of packages that contain 3 ounces or more of chocolate stars) / (total number of packages)
x̄ = 42 / 50 x̄ = 0.84
The p-value is the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true. In this case, we are interested in the probability of observing a test statistic greater than 0.84, assuming that the null hypothesis is true.
Using a standard normal distribution table or calculator, we can find the p-value:
p-value = P(Z > 0.84) p-value = 0.1995
Based on the p-value, we can conclude that the null hypothesis is not rejected. This means that we fail to reject the null hypothesis that the proportion of packages that contain 3 ounces or more of chocolate stars is less than or equal to 80%.
However, we must note that the p-value is relatively high (0.1995), which means that the test is not very powerful. This is because the sample size is relatively small (n = 50), and the standard deviation of the sampling distribution is relatively large.
The implications of this study are that the candy company may not be able to claim that the bridge mix is mostly chocolate stars, based on the data from the quality control test. However, it is essential to note that this conclusion is based on a relatively small sample size, and the results may not be generalizable to the entire population of bridge mix packages.
To improve the accuracy of the results, the candy company could consider increasing the sample size or using a more robust statistical method, such as a confidence interval. Additionally, the company could consider collecting more data on the weight of chocolate stars in each package to improve the precision of the estimates.
In conclusion, the quality control test on a random sample of 50 packages suggests that the proportion of packages that contain 3 ounces or more of chocolate stars may be less than 80%. However, the results are not conclusive, and further research is needed to confirm the findings.
Frequently Asked Questions (FAQs) About the Candy Company's Bridge Mix
A: The purpose of the quality control test is to determine if the proportion of packages that contain 3 ounces or more of chocolate stars in the bridge mix is at least 80%. This is a regulatory requirement for the candy company to claim that the bridge mix is mostly chocolate stars.
A: The null hypothesis is that the proportion of packages that contain 3 ounces or more of chocolate stars is less than or equal to 80% (H0: p ≤ 0.8).
A: The alternative hypothesis is that the proportion of packages that contain 3 ounces or more of chocolate stars is greater than 80% (H1: p > 0.8).
A: The test statistic is the proportion of packages that contain 3 ounces or more of chocolate stars, which is calculated as x̄ = 42 / 50 = 0.84.
A: The p-value is the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true. In this case, the p-value is 0.1995.
A: The p-value indicates that we fail to reject the null hypothesis that the proportion of packages that contain 3 ounces or more of chocolate stars is less than or equal to 80%.
A: The implications of the study are that the candy company may not be able to claim that the bridge mix is mostly chocolate stars, based on the data from the quality control test. However, it is essential to note that this conclusion is based on a relatively small sample size, and the results may not be generalizable to the entire population of bridge mix packages.
A: Some potential limitations of the study include:
- The sample size is relatively small (n = 50), which may not be representative of the entire population of bridge mix packages.
- The standard deviation of the sampling distribution is relatively large, which may affect the accuracy of the results.
- The study only examines the proportion of packages that contain 3 ounces or more of chocolate stars, and does not consider other factors that may affect the quality of the bridge mix.
A: Some potential future directions for the study include:
- Increasing the sample size to improve the accuracy of the results.
- Using a more robust statistical method, such as a confidence interval, to improve the precision of the estimates.
- Collecting more data on the weight of chocolate stars in each package to improve the precision of the estimates.
- Examining other factors that may affect the quality of the bridge mix, such as the type of chocolate used or the manufacturing process.
A: The next step for the candy company is to consider the implications of the study and determine the best course of action. This may involve increasing the sample size, using a more robust statistical method, or collecting more data on the weight of chocolate stars in each package. Ultimately, the goal is to ensure that the bridge mix meets the required standards and is safe for consumption.