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Chips per Bag In a 1998 advertising campaign, Nabisco claimed that every 18-ounce bag of Chips Ahoy! cookies contained at least 1000 chocolate chips. Brad Warner and Jim Rutledge tried to verify the claim. The following data represent the number of chips in an 18 -ounce bag of Chips Ahoy! based on their study. $$ \begin{array}{lllll} \hline 1087 & 1098 & 1103 & 1121 & 1132 \\ \hline 1185 & 1191 & 1199 & 1200 & 1213 \\ \hline 1239 & 1244 & 1247 & 1258 & 1269 \\ \hline 1307 & 1325 & 1345 & 1356 & 1363 \\ \hline 1135 & 1137 & 1143 & 1154 & 1166 \\ \hline 1214 & 1215 & 1219 & 1219 & 1228 \\ \hline 1270 & 1279 & 1293 & 1294 & 1295 \\ \hline 1377 & 1402 & 1419 & 1440 & 1514 \\ \hline \end{array} $$ (a) Draw a normal probability plot to determine if the data could have come from a normal distribution. (b) Determine the mean and standard deviation of the sample data. (c) Using the sample mean and sample standard deviation obtained in part (b) as estimates for the population mean and population standard deviation, respectively, draw a graph of a normal model for the distribution of chips in a bag of Chips Ahoy! (d) Using the normal model from part (c), find the probability that an 18-ounce bag of Chips Ahoy! selected at random contains at least 1000 chips. (e) Using the normal model from part (c), determine the proportion of 18 -ounce bags of Chips Ahoy! that contains between 1200 and 1400 chips, inclusive.

Step by step solution

01

Construct a Normal Probability Plot

Using the given data set, plot the number of chips against the expected z-scores. If the points lie approximately along a straight line, the data may be considered as coming from a normal distribution.

02

Calculate Mean and Standard Deviation

Compute the mean \(\bar{x}\) and standard deviation \(s\) of the sample. Use the following formulas: \[ \bar{x} = \frac{1}{N} \sum_{i=1}^N x_i \] \[ s = \sqrt{ \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2 } \] where \(N\) is the number of samples, and \(x_i\) are the individual data points.

03

Graph the Normal Model

Using the calculated sample mean and standard deviation, graph the normal distribution curve. The mean represents the center of the distribution, and the standard deviation controls the spread.

04

Find Probability for At Least 1000 Chips

Utilize the standard normal distribution table or a calculator to find the probability that the count of chips is at least 1000. Compute the z-score: \[ z = \frac{x - \mu}{\sigma} \] where \(x = 1000\), \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Find the area to the right of this z-score.

05

Determine Proportion Between 1200 and 1400 Chips

Compute the z-scores for 1200 and 1400 chips. Find the areas corresponding to these z-scores using the standard normal distribution table or a calculator. The proportion is the difference between these areas. \[ z_{1200} = \frac{1200 - \mu}{\sigma} \] \[ z_{1400} = \frac{1400 - \mu}{\sigma} \] The proportion is \[ P(1200 \leq X \leq 1400) = \Phi(z_{1400}) - \Phi(z_{1200}) \]

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Key Concepts

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

mean and standard deviation

Understanding the mean and standard deviation is crucial for analyzing any set of data, including the number of chocolate chips in a bag of Chips Ahoy! cookies. The mean, represented as \( \bar{x} \), is the average of all data points. It gives us a central value around which the data points tend to cluster. To find the mean, sum up all the individual data points and then divide by the total number of points, \( N \). The formula is: \[ \bar{x} = \frac{1}{N} \bigg( \text{sum of all data points} \bigg) \]
The standard deviation, denoted by \( s \), measures the spread or dispersion of the data points around the mean. A smaller standard deviation indicates that the data points are close to the mean, while a larger standard deviation shows that the points are more spread out. The formula to calculate the standard deviation is: \[ s = \frac{1}{N-1} \bigg( \text{sum of squared differences between each data point and the mean} \bigg)^{1/2} \]
Both these statistics are essential for drawing a normal distribution, which helps us understand the probabilities of different outcomes within our data set.

normal probability plot

A normal probability plot helps us determine if the data follows a normal distribution. For the Chips Ahoy! data, constructing this plot involves plotting the given number of chips against the expected z-scores.
In a normal probability plot, the x-axis typically represents the expected z-scores, and the y-axis represents the observed data. If the points on the plot fall approximately along a straight line, it indicates that the data likely follows a normal distribution.
This method is visual and relatively simple, making it easier for students to identify deviations from normality. Understanding whether data is normally distributed is important because it justifies using statistical methods that assume normality, such as z-scores and standard normal distribution tables in subsequent analyses.

z-scores

Z-scores are a way to standardize data points within the context of a normal distribution. When we calculate a z-score, we are determining how many standard deviations a data point is from the mean.
The formula for finding a z-score is: \[ z = \frac{x - \bar{x}}{s} \]
Here, \( x \) is the data point, \( \bar{x} \) is the mean, and \( s \) is the standard deviation. For instance, if we want to find the probability that a bag of Chips Ahoy! contains at least 1000 chips, we first compute its z-score: \[ z = \frac{1000 - \bar{x}}{s} \]
After computing the z-score, we can use standard normal distribution tables or a calculator to find the corresponding probability. This is the area under the curve to the right of the z-score if we are interested in 'at least' scenarios, or between two z-scores for range scenarios. For example, if we want to find the proportion of bags having between 1200 and 1400 chips, we calculate z-scores for both 1200 and 1400, then find the area between these z-scores in the standard normal distribution table: \[ z_{1200} = \frac{1200 - \bar{x}}{s} \] \[ z_{1400} = \frac{1400 - \bar{x}}{s} \]
The proportion is derived by subtracting the area below \( z_{1200} \) from the area below \( z_{1400} \). Understanding z-scores is fundamental in statistics, as they allow for comparisons between different data sets and facilitate probability calculations.