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3 years ago

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Mr. Jimerson records the amount of snowfall that the local area recieves. 12am-1am: 0.3 inches | 1am-2am: 0.8 inches | 2am-3am:

1.36 inches |3am-4am: 2.019. If the total amount of snowfall at 7am is 7.8 inches. How much snow fell between 4am-7am? hurry pls

1 answer:

MatroZZZ [7]3 years ago

6 0

Answer:

5.781 inches.

Step-by-step explanation:

Since Mr. Jimerson records the amount of snowfall that the local area recieves, recording the following data: 12 am-1am: 0.3 inches | 1 am-2am: 0.8 inches | 2 am-3am: 1.36 inches | 3 am-4am: 2.019, if the total amount of snowfall at 7am is 7.8 inches, to determine how much snow fell between 4 am-7am the following calculation must be performed:

7.8 - 2.019 = X

5.781 = X

Therefore, since the amount of snow accumulated at 7 am was 7.8 inches, the snowfall between 4 and 7 am was 5.781 inches.

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Step-by-step explanation:

To solve this problem, it is important to know the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

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The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$ , a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 11, \sigma = 3$

a. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?

Here we have that $n = 25, s = \frac{3}{\sqrt{25}} = 0.6$

This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.

X = 11.2

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{11.2 - 11}{0.6}$

$Z = 0.33$

$Z = 0.33$ has a pvalue of 0.6293.

X = 10.8

$Z = \frac{X - \mu}{s}$

$Z = \frac{10.8 - 11}{0.6}$

$Z = -0.33$

$Z = -0.33$ has a pvalue of 0.3707.

0.6293 - 0.3707 = 0.2586

0.259 probability, rounded to three decimal places.

b. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.5 and 11 minutes?

Subtraction of the pvalue of Z when X = 11 subtracted by the pvalue of Z when X = 10.5. So

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$Z = \frac{11 - 11}{0.6}$

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$Z = 0$ has a pvalue of 0.5.

X = 10.5

$Z = \frac{X - \mu}{s}$

$Z = \frac{10.5 - 11}{0.6}$

$Z = -0.83$

$Z = -0.83$ has a pvalue of 0.2033.

0.5 - 0.2033 = 0.2967

0.297, rounded to three decimal places.

c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?

Here we have that $n = 100, s = \frac{3}{\sqrt{100}} = 0.3$

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X = 11.2

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0.7486 - 0.2514 = 0.4972

0.497, rounded to three decimal places.

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