The basal diameter of a sea anemone is an indicator of its age, and in a certain population of anemones, the distribution of basal diameters is approximately normal with a mean of 5.3 cm and a standard deviation of 1.8 cm. suppose you randomly select five anemones from this population.
a. what is the probability that all five anemones have a basal diameter more than 5.5 cm? (2pt)

The probability that all five randomly selected sea anemones from a population with normally distributed basal diameters have a diameter more than 5.5 cm is approximately 4.5%.

To solve this problem, we first need to find the probability that one sea anemone has a basal diameter more than 5.5 cm, given the population's mean basal diameter is 5.3 cm with a standard deviation of 1.8 cm. We calculate the Z-score for a diameter of 5.5 cm to see how many standard deviations this value is away from the mean. The Z-score formula is Z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation.

The Z-score for a diameter of 5.5 cm is Z = (5.5 - 5.3) / 1.8 ≈ 0.11. Looking this up in a standard normal distribution table, we find that the probability of selecting an anemone with a diameter larger than 5.5 cm is approximately 0.5438. Since the selections are independent, the probability of all five having a diameter more than 5.5 cm is (0.5438)⁵ ≈ 0.045, or 4.5%.

The probability that all five randomly selected anemones have a basal diameter more than 5.5 cm is approximately 0.0056.

To solve this problem, we can use the standard normal distribution. First, we need to standardize the diameter value using the z-score formula:

[tex]z = \frac{x - \mu}{\sigma}[/tex]

where:

- x is the value of the basal diameter,

- [tex]\mu[/tex] is the mean of the basal diameters (5.3 cm),

- [tex]\sigma[/tex] is the standard deviation of the basal diameters (1.8 cm), and

- z is the standardized score.

For a basal diameter of 5.5 cm, the z-score would be:

[tex]z = \frac{5.5 - 5.3}{1.8} = \frac{0.2}{1.8} \approx 0.1111[/tex]

To find the probability that an anemone has a basal diameter more than 5.5 cm, we need to find the area to the right of z = 0.1111 under the standard normal curve.

Using a standard normal table or calculator, we find that this area is approximately 0.4562.

Since each selection is independent (assuming replacement), the probability that all five anemones have a basal diameter more than 5.5 cm is the probability raised to the power of 5:

[tex]P(\text{all five} > \text{ 5.5 cm}) = (0.4562)^5 \approx 0.0056[/tex]

So, the probability that all five randomly selected anemones have a basal diameter more than 5.5 cm is approximately 0.0056.

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