To find the probability that a randomly selected bag contains between 1100 and 1400 chocolate chips, we can use the properties of the normal distribution. Here are the steps:

Step 1: Standardize the values (convert to z-scores)

The z-score for a value $x$ is calculated as:

$z = \frac{x - \mu}{\sigma}$

where:

• $\mu = 1252$ (the mean number of chips),
• $\sigma = 129$ (the standard deviation),
• $x$ is the value for which we want to find the z-score.

We need to calculate the z-scores for $x = 1100$ and $x = 1400$.

For $x = 1100$:

$z_{1100} = \frac{1100 - 1252}{129} = \frac{-152}{129} \approx -1.18$

For $x = 1400$:

$z_{1400} = \frac{1400 - 1252}{129} = \frac{148}{129} \approx 1.15$

Step 2: Use the z-table to find probabilities

Now that we have the z-scores, we look up the corresponding probabilities for $z = -1.18$ and $z = 1.15$ in the standard normal distribution table (z-table) or use a calculator.

• The probability corresponding to $z = -1.18$ is approximately 0.1190.
• The probability corresponding to $z = 1.15$ is approximately 0.8749.

Step 3: Find the probability between 1100 and 1400

The probability that a randomly selected bag contains between 1100 and 1400 chocolate chips is the difference between the two probabilities:

$P(1100 < X < 1400) = P(Z = 1.15) - P(Z = -1.18)$

$P(1100 < X < 1400) = 0.8749 - 0.1190 = 0.7559$

Thus, the probability is approximately 0.7559 or 75.59%.

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Would you like further explanation on any of the steps, or do you have additional questions? Here are some related topics you might find helpful:

1. How to calculate z-scores for different distributions.
2. The interpretation of the normal distribution curve.
3. How to use a standard normal table (z-table).
4. Why the normal distribution is important in statistics.
5. How to calculate probabilities for values outside the mean in a normal distribution.

Tip: The z-score tells you how many standard deviations a particular value is from the mean.