91Ó°ÊÓ

Chips Ahoy In \(1998,\) as an advertising campaign, the Nabisco Company announced a " 1000 Chips Challenge," claiming that every 18 -ounce bag of their Chips Ahoy cookies contained at least 1000 chocolate chips. Dedicated Statistics students at the Air Force Academy (no kidding) purchased some randomly selected bags of cookies, and counted the chocolate chips. Some of their data are given below. (Chance, \(12,\) no. 1[1999] ) $$\begin{array}{llllllll} 1219 & 1214 & 1087 & 1200 & 1419 & 1121 & 1325 & 1345 \\ 1244 & 1258 & 1356 & 1132 & 1191 & 1270 & 1295 & 1135 \end{array}$$ a) Check the assumptions and conditions for inference. Comment on any concerns you have. b) Create a \(95 \%\) confidence interval for the average number of chips in bags of Chips Ahoy cookies. c) What does this evidence say about Nabisco's claim? Use your confidence interval to test an appropriate hypothesis and state your conclusion.

Step by step solution

01

Check Assumptions and Conditions

Before performing any statistical inference, we must check if our data meets the necessary conditions for the analysis. These conditions include: 1. **Randomization**: The bags were randomly selected, as per the problem statement. 2. **Independence**: Each observed chip count is assumed to be independent of the others, typical for random samples taken from a large population. 3. **Sample Size/Normality**: The sample size here is 16. With a smaller sample size (less than 30), we usually assume normality of the sampling distribution, or that the data itself is approximately normally distributed. We can make a quick visual inspection or use statistical tests to confirm this. Assuming these conditions are met, we can proceed with inference. If normality is a concern, transformation or bootstrapping could be considered.

02

Calculate the Sample Mean and Standard Deviation

We need to calculate the sample mean \(\bar{x}\) and the sample standard deviation \(s\) for the data given. Let’s calculate:- Sample Mean \(\bar{x} = \frac{1219 + 1214 + \ldots + 1135}{16} = 1240.25\)- Compute each deviation from the mean, square it, and find variance \[ s^2 = \frac{(1219-1240.25)^2 + (1214-1240.25)^2 + \ldots + (1135-1240.25)^2}{15} \] Hence, \[ s \approx 98.22 \]

03

Create a 95% Confidence Interval

For a 95% confidence interval of the mean, given a small sample size, we use the t-distribution. The formula for the confidence interval is:\[ \bar{x} \pm t^* \frac{s}{\sqrt{n}} \]Where \(t^*\) is the t-value from the t-distribution table corresponding to 95% confidence and \(n-1\) degrees of freedom (which is 15 in this case).Assuming \(t^*\approx 2.131\):- Plug in the values: \[ 1240.25 \pm 2.131 \frac{98.22}{\sqrt{16}} = 1240.25 \pm 52.09 \]Resulting in a confidence interval of \([1188.16, 1292.34]\).

04

Conclusion about Nabisco's Claim

The 95% confidence interval \([1188.16, 1292.34]\) does not include the number 1000, which means we are fairly confident that the true average number of chips per bag is indeed greater than 1000.According to the confidence interval, Nabisco's claim that every bag has at least 1000 chips holds true based on this data, as the entire interval is above 1000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

Statistical inference is a powerful tool, allowing us to make predictions about a population based on sample data. One of its key concepts is the confidence interval. A confidence interval provides a range of values which is likely to contain the true population parameter. In this exercise, we created a 95% confidence interval to estimate the average number of chocolate chips in a bag of Chips Ahoy cookies.

Here’s how we calculated it:

• First, we found the sample mean, \(\bar{x} = 1240.25\).
• Next, we used the sample standard deviation, \(s = 98.22\), to measure the variation of chips in the bags.
• We then applied the t-distribution formula for small sample sizes, given as: \[\bar{x} \pm t^* \frac{s}{\sqrt{n}}\]\
• Using a t-value of approximately 2.131 for a 95% confidence level with 15 degrees of freedom, we plug the values into the formula.
• This produced an interval between 1188.16 and 1292.34 chips.

This interval tells us we can be 95% confident that the average number of chips in all Chips Ahoy bags will fall within this range, supporting Nabisco's claim that bags have at least 1000 chocolate chips.

Hypothesis Testing

Hypothesis testing is another cornerstone of statistical inference. It helps us decide if there is enough evidence to support a specific claim about a population parameter.

For the Nabisco case, we wanted to test the hypothesis that each bag of Chips Ahoy contains at least 1000 chips. Let's break down how we approached it:

• We started with the null hypothesis, \(H_0\): the average number of chips per bag is at least 1000.
• The alternative hypothesis, \(H_a\), was that the average might be less than 1000.
• Using the 95% confidence interval we calculated (1188.16 to 1292.34), we observed it did not include 1000.
• Because the entire interval is above 1000, we can infer there's enough evidence to reject the null hypothesis.

Hence, the hypothesis test supports Nabisco's claim, strengthening our confidence in it while also visibly presenting the data's implications.

Assumptions and Conditions

Before performing statistical analysis, checking assumptions and conditions is essential to ensure reliable results. For this type of inference, several assumptions and conditions should be met:

• **Randomization:** The sample of cookie bags was randomly selected, which enhances the representativeness of the data.
• **Independence:** Each observation, i.e., the count of chips in each bag, should be independent of others. With random sampling from a large population, this condition is typically fulfilled.
• **Sample Size/Normality:** For smaller samples, such as our 16 bags, we assume the distribution of sample means is approximately normal. If normality appears skewed through quick visual inspection or tests, transformations or bootstrapping might be necessary.

Meeting these conditions allows us to proceed with confidence in our analysis, as they validate the reliability of our results.

t-Distribution

The t-distribution is an essential concept in statistical inference, particularly useful when working with small sample sizes. Unlike the normal distribution, the t-distribution accounts for extra variability by having heavier tails. This makes it robust for intense parameter estimation from small samples.

Why is the t-distribution significant here?

• When creating a confidence interval for the mean number of chocolate chips based on only 16 samples, we use the t-distribution because the population standard deviation is unknown.
• It adjusts the confidence interval to maintain its accuracy despite the smaller sample size.
• The t-value (about 2.131 in our exercise) varies with sample size; specifically, `n-1` degrees of freedom influence its estimation from statistical tables.

Hence, the choice of the t-distribution rather than the normal distribution ensured the precision and reliability of our inference about the average number of chips in the bags.