Let’s solve each part step by step:

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a) Percentage of cookies weighing more than 68 grams

The weights of the cookies follow a normal distribution with:

$\mu = 75 \, \text{grams}, \, \sigma = 8 \, \text{grams}$

We need to find $P(X > 68)$, where $X$ is the weight of a single cookie.

1. Standardize $X$: $Z = \frac{X - \mu}{\sigma}$ Substituting $X = 68$: $Z = \frac{68 - 75}{8} = \frac{-7}{8} = -0.875$

2. Find the probability: Using a standard normal table or calculator, find $P(Z > -0.875)$. For symmetry, we use: $P(Z > -0.875) = 1 - P(Z \leq -0.875)$

   From standard normal distribution tables: $P(Z \leq -0.875) \approx 0.1908$

   So: $P(Z > -0.875) = 1 - 0.1908 = 0.8092$

   Therefore, 80.92% of the cookies weigh more than 68 grams.

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b) Probability that the total weight of cookies in a package is less than 900 grams

The total weight of 13 cookies is the sum of 13 independent normally distributed random variables. The sum also follows a normal distribution with:

$\text{Mean of total weight: } \mu_{\text{total}} = n \cdot \mu = 13 \cdot 75 = 975 \, \text{grams}$ $\text{Standard deviation of total weight: } \sigma_{\text{total}} = \sqrt{n} \cdot \sigma = \sqrt{13} \cdot 8 \approx 28.844 \, \text{grams}$

We need $P(\text{Total Weight} < 900)$, or $P(Y < 900)$, where $Y$ is the total weight.

1. Standardize $Y$: $Z = \frac{Y - \mu_{\text{total}}}{\sigma_{\text{total}}}$ Substituting $Y = 900$: $Z = \frac{900 - 975}{28.844} = \frac{-75}{28.844} \approx -2.600$

2. Find the probability: From standard normal distribution tables: $P(Z < -2.600) \approx 0.0047$

   Therefore, the probability is approximately 0.0047 (0.47%).

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Final Answers:

a) 80.920
b) 0.005

Let me know if you want further clarifications or a step-by-step explanation of any part!

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Related Questions:

1. What is the probability that a single cookie weighs less than 60 grams?
2. How does increasing the sample size affect the standard deviation of the total weight?
3. What percentage of the cookies weigh between 70 and 80 grams?
4. What is the z-score for a cookie weighing 85 grams?
5. What assumptions are made about the distribution of cookie weights?

Tip:

To quickly compute probabilities for normal distributions, tools like z-tables, calculators, or software like Excel can be extremely helpful!