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The mass of northern pike in Candle Lake is normally distributed with a mean of 3.5 kg and a standard deviation of 0.6 kg.

If during the course of the fishing season you catch 135 northern pike at Candle Lake, how many of these are likely to have a mass that is more than 5.3 kg? Round your answer to the nearest whole number.

Answer :

To find out how many northern pike are likely to weigh more than 5.3 kg, we need to calculate the probability of catching a pike that weighs more than this amount and then apply this probability to the total number caught, which is 135.

First, we need to determine how far 5.3 kg is from the mean in terms of standard deviations. This is done using the z-score formula:

[tex]z = \frac{X - \text{mean}}{\text{standard deviation}}[/tex]

Where:

  • [tex]X[/tex] is the value of interest (5.3 kg),
  • The mean is 3.5 kg,
  • The standard deviation is 0.6 kg.

[tex]z = \frac{5.3 - 3.5}{0.6} = \frac{1.8}{0.6} = 3[/tex]

A z-score of 3 means 5.3 kg is 3 standard deviations above the mean.

Next, we use the standard normal distribution table (or a calculator) to find the probability that a randomly caught fish weighs more than 5.3 kg. For a z-score of 3, the probability of a fish weighing less than 5.3 kg is approximately 0.9987.

Therefore, the probability of a fish weighing more than 5.3 kg is:

[tex]P(X > 5.3) = 1 - P(X < 5.3) = 1 - 0.9987 = 0.0013[/tex]

To find the expected number of fish over this weight, multiply this probability by the total number of fish caught:

[tex]0.0013 \times 135 = 0.1755[/tex]

Rounding this number to the nearest whole number gives us approximately 0 fish. Therefore, it is likely that very few, if any, of the 135 captured northern pike would weigh more than 5.3 kg.

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