Let X be a standard normal random variable so that X∼N(0,1). For this problem you may want to refer to the table provided on Canvas. Recall, that Φ(x) denotes the standard normal CDF.(a) Find Φ(1.45).(b) Find x, such that Φ(x) = 0.4.(c) Based on the fact that Φ(1.645) = 0.95 find an interval in which X will fall with 95% probability.(d) Find another interval (different from the one in (c)) into which X will fall with 95% probability. Hint you may want to consider the symmetric properties of the standard normaldistribution.(e) Find an interval with lower bound -1.9 into which X will fall with 95% probability.(f) Find an interval with upper bound 2.1 into which X will fall with 95% probability.(g) Among all possible intervals into which X falls with 95% probability, find the shortest one.

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Asked Feb 5, 2020
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Let X be a standard normal random variable so that X∼N(0,1). For this problem you may want to refer to the table provided on Canvas. Recall, that Φ(x) denotes the standard normal CDF.
(a) Find Φ(1.45).
(b) Find x, such that Φ(x) = 0.4.
(c) Based on the fact that Φ(1.645) = 0.95 find an interval in which X will fall with 95% probability.
(d) Find another interval (different from the one in (c)) into which X will fall with 95% probability. Hint you may want to consider the symmetric properties of the standard normal
distribution.
(e) Find an interval with lower bound -1.9 into which X will fall with 95% probability.
(f) Find an interval with upper bound 2.1 into which X will fall with 95% probability.
(g) Among all possible intervals into which X falls with 95% probability, find the shortest one.
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Expert Answer

Step 1

Note:

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Since the normal table is not given, we have calculated the probability value using Excel.

Step 2

The normal distribution:

A continuous random variable X is said to follow normal distribution with mean µ and standard deviation σ if the probability density function of X is,

Statistics homework question answer, step 2, image 1

Step 3

a)

Here, X is a standard normal variable.

The required probability is calculated as follows:

Statistics homework question answer, step 3, image 1

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