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Helga [31]
3 years ago
13

Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model, with a

growth rate parameter of 17% per hour. Suppose also that a sample culture of 1600 is obtained from this population. Find the size of the sample after four hours. Round your awnser to the nearest integer
Mathematics
1 answer:
Len [333]3 years ago
4 0

Answer:

The size of the sample after four hours is of 3158 bacteria.

Step-by-step explanation:

The population after t hours can be modeled by the continuous exponential growth model, which is:

P(t) = P(0)e^{rt}

In which P(0) is the initial population and r is the growth rate paremeter.

A growth rate parameter of 17% per hour.

This means that r = 0.17

Suppose also that a sample culture of 1600 is obtained from this population.

This means that P(0) = 1600

Find the size of the sample after four hours.

This is P(4).

P(t) = P(0)e^{rt}

P(t) = 1600e^{0.17t}

P(4) = 1600e^{0.17*4}

P(4) = 3158.2

Rounding to the nearest integer

The size of the sample after four hours is of 3158 bacteria.

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Answer:

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Step-by-step explanation:

To solve this question, we need to understand 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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question, we have that:

\mu = 245 \sigma = 15, n = 33, s = \frac{15}{\sqrt{33}} = 2.61

What the probability that the sample mean will be 246 pages or more?

This is 1 subtracted by the pvalue of Z when X = 246. So

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

By the Central Limit Theorem

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

Z = \frac{246 - 245}{2.61}

Z = 0.38

Z = 0.38 has a pvalue of 0.6480.

1 - 0.6480 = 0.3520

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