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zhenek [66]
3 years ago
13

Use technology or a z-score table to answer the question.

Mathematics
1 answer:
Alik [6]3 years ago
3 0

Answer:

The second choice: Approximately 65.2\% of the pretzel bags here will contain between 225 and 245 pretzels.

Step-by-step explanation:

This explanation uses a z-score table where each z entry has two decimal places.

Let \mu represent the mean of a normal distribution of variable X. Let \sigma be the standard deviation of the distribution. The z-score for the observation x would be:

\displaystyle z = \frac{x - \mu}{\sigma}.

In this question,

  • \mu = 240.
  • \sigma = 9.3.

Calculate the z-score for x_1 = 225 and x_2 = 245. Keep in mind that each entry in the z-score table here has two decimal places. Hence, round the results below so that each contains at least two decimal places.

\begin{aligned} z_1 &= \frac{x_1 - \mu}{\sigma} \\ &= \frac{225 - 240}{9.3} \approx -1.61\end{aligned}.

\begin{aligned} z_2 &= \frac{x_2 - \mu}{\sigma} \\ &= \frac{245 - 240}{9.3} \approx 0.54\end{aligned}.

The question is asking for the probability P(225 \le X \le 245) (where X is between two values.) In this case, that's the same as P(-1.61 \le Z \le 0.54).

Keep in mind that the probabilities on many z-table correspond to probability of P(Z \le z) (where Z is no greater than one value.) Therefore, apply the identity P(z_1 \le Z \le z_2) = P(Z \le z_2) - P(Z \le z_1) to rewrite P(-1.61 \le Z \le 0.54) as the difference between two probabilities:

P(-1.61 \le Z \le 0.54) = P(Z \le 0.54) - P(Z \le -1.61).

Look up the z-table for P(Z \le 0.54) and P(Z \le -1.61):

  • P(Z \le 0.54)\approx 0.70540.
  • P(Z \le -1.61) \approx 0.05370.

\begin{aligned}& P(225 \le X \le 245) \\ &= P\left(\frac{225 - 240}{9.3} \le Z \le \frac{245 - 240}{9.3}\right)\\&\approx P(-1.61 \le Z \le 0.54) \\ &= P(Z \le 0.54) - P(Z \le -1.61)\\ &\approx 0.70540 - 0.05370 \\& \approx 0.65.2 \\ &= 65.2\% \end{aligned}.

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Answer:

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Step-by-step explanation:

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