The distances that aeroplanes move before coming to a stop after landing have been observed to be normally distributed with a mean of [tex]\mu[/tex] metres and standard deviation of [tex]\sigma[/tex] metres. If it is known that 8% of aeroplanes move a distance of at least 1140 metres before coming to a stop, while 90% of aeroplanes travel up to 1100 metres before coming to a stop, then [tex]\mu[/tex] is:

Given that 8% of airplanes move a distance of at least 1140 meters before coming to a stop, we can infer that this distance is 1.28 standard deviations above the mean (assuming a normal distribution). Similarly, since 90% of airplanes travel up to 1100 meters before coming to a stop, this distance is 0.89 standard deviations below the mean.

To find the mean distance traveled by airplanes, we can use the properties of the normal distribution. From the given information, we know that the distance of at least 1140 meters is 1.28 standard deviations above the mean, and the distance up to 1100 meters is 0.89 standard deviations below the mean. Using these values, we can set up two equations:

1.28σ + μ = 1140 (equation 1)

-0.89σ + μ = 1100 (equation 2)

Solving these equations simultaneously, we can find the value of μ (the mean):

Adding equation 1 and equation 2:

1.28σ + μ + (-0.89σ + μ) = 1140 + 1100

0.39σ + 2μ = 2240

Dividing both sides by 2:

0.39σ/2 + 2μ/2 = 2240/2

0.195σ + μ = 1120

Substituting equation 1 into the above equation:

1140 + μ = 1120

μ = 1120 - 1140

μ = -20

Therefore, the mean distance traveled by airplanes before coming to a stop after landing is -20 meters.

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