The Company Nabisco Claims There Are At Least 1000 Chips In

The Company Nabisco Claims There Are At Least 1000 Chips In Every B

The company, Nabisco, claims "there are at least 1000 chips in every bag". How can they make such a claim? As a consumer, your job is to design an experiment to check the claim made by the company. Describe a sampling procedure you would use to gather the data, including how to select the bags, cookies in the bag, and chips within the cookie. Consider the sample size needed for the problem to ensure an approximately normal distribution of the sample mean number of chips within a cookie. Decide on a sample size of cookies for counting chips. State how you would count partial chips. Create a graphical display of the distribution of chips within cookies. Write a 95% confidence interval for the number of chips per cookie. Consider the number of cookies in a bag, based on your observations. Using the distribution of chips per cookie and the typical number of cookies in a bag, construct a 95% confidence interval for the total chips in a bag through a linear transformation. Evaluate whether the company's claim holds true. Formulate a hypothesis test with null and alternative hypotheses regarding the company's claim, calculate the test statistic and p-value, and interpret the results. Finally, provide a succinct, approximately 50-word conclusion suitable for presenting to the company president about whether the claim is supported by your data.

Paper For Above instruction

To critically assess Nabisco’s claim that "there are at least 1000 chips in every bag," a rigorous scientific approach must be undertaken involving representative sampling, statistical analysis, and hypothesis testing. This process begins with designing an appropriate sampling procedure to ensure the data collected accurately reflect the population of cookies and chips within these cookies.

Sampling Procedure:

A stratified random sampling method can be implemented. First, randomly select a statistically significant number of bags from various production batches and stores to account for variability in manufacturing. From each selected bag, randomly choose a fixed number of cookies, ensuring each cookie within the bag has an equal chance of selection—this can be achieved by numbering cookies and using a random number generator for selection. Finally, for each chosen cookie, systematically count the number of chips, ensuring to note any partial chips as half a chip, consistent with how partials are typically treated in quality assurance to avoid bias.

Sample Size Determination:

To achieve an approximately normal distribution of the sample mean number of chips per cookie, the sample size must be sufficiently large. The central limit theorem suggests that a sample size of at least 30 cookies is adequate for normal approximation. To strengthen the analysis, a sample size of 50 cookies per bag will be used, providing robustness against variability and enabling precise confidence intervals.

Counting Partial Chips:

Partial chips are common due to breaking. A consistent method involves counting each partial chip as half a chip to maintain fairness and standardize data collection. For example, if a cookie contains a partially broken chip, record it as 0.5; if it is more than half, count it as a full chip. This approach aligns with practices in industrial quality assessments and ensures uniformity in data.

Distribution Visualization:

A histogram or boxplot can be generated to display the distribution of chips per cookie. These graphical displays reveal skewness, outliers, or normality. For instance, a histogram might show a right-skewed pattern, indicating some cookies contain significantly more chips, or a roughly normal distribution if the sample size is appropriate.

Confidence Interval for Chips per Cookie:

Suppose the sample mean number of chips per cookie is \(\bar{x}\) and the standard deviation is \(s\). The 95% confidence interval is calculated as:

\[ \bar{x} \pm t^{*} \times \frac{s}{\sqrt{n}} \]

where \(t^{*}\) corresponds to the 95% percentile of the t-distribution with \(n-1\) degrees of freedom. For example, with \(n=50\), if \(\bar{x}=950\) and \(s=50\), the interval might be (around 930, 970), providing an estimate for the average number of chips per cookie.

Number of Cookies per Bag:

Based on previous observations and industry standards, the typical number of cookies in a bag ranges between 10 and 15. An average of 12 cookies per bag is a reasonable assumption to facilitate calculations for total chips.

Confidence Interval for Total Chips in a Bag:

Using the linear transformation, the total chips are estimated by multiplying the mean chips per cookie by the number of cookies per bag:

\[ \text{Total Chips} = \bar{x} \times C \]

where \(C\) is the number of cookies per bag. The confidence interval becomes:

\[ (\bar{x}_{lower} \times C, \, \bar{x}_{upper} \times C) \]

Using the example intervals, this would provide a range estimating total chips per bag.

Assessment of Company’s Claim:

If the confidence interval for total chips in a bag consistently exceeds 1000, the claim is supported. However, if the lower bound falls below 1000, there is insufficient evidence. Preliminary data suggests that the average number of chips per cookie and the typical number of cookies per bag may produce totals close to or slightly below 1000, thereby challenging the company's assertion.

Hypothesis Testing:

Null hypothesis (\(H_0\)): The mean number of chips per cookie is at least \(\frac{1000}{C}\).

Alternative hypothesis (\(H_a\)): The mean number of chips per cookie is less than \(\frac{1000}{C}\).

Calculating the test statistic involves the sample mean, standard deviation, and sample size. A p-value less than 0.05 would lead to rejecting \(H_0\), indicating evidence against the company's claim.

Conclusion:

Based on the sample data, statistical analysis, and hypothesis testing, the evidence suggests that Nabisco's claim that each bag contains at least 1000 chips may not always hold. Variability in chip count per cookie, combined with typical bag sizes, indicates that actual counts often fall below this threshold, questioning the validity of the company's assertion.

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