PRACTICE EXERCISE 7-1

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Problem 1. Find the area under the normal curve to the left of \(z = 1.34\).

round(pnorm(1.34, lower.tail = TRUE), digits = 4)

[1] 0.9099

The area under the normal curve to the left of \(z=1.34\) is \(0.9099\).

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Problem 2. Find the area under the normal curve to the right of \(z=-0.52\).

round(pnorm(-0.52, lower.tail = FALSE), digits = 4)

[1] 0.6985

The area under the normal curve to the right of \(z=-0.52\) is \(0.6985\).

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Problem 3. Find the area between \(z=-1.3\) and \(z=2.4\).

round(pnorm(2.4)-pnorm(-1.3), digits = 4)

[1] 0.895

The area between \(z=-1.3\) and \(z=2.4\) is \(0.895\).

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Problem 4. Find the \(z\)-score when the area to the left of it is \(0.8621\).

round(qnorm(0.8621, lower.tail = TRUE), digits = 2)

[1] 1.09

The \(z\)-score with an area of \(0.8621\) to its left is \(1.09\)

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Problem 5. Find the \(z\)-score when the area to the right of \(z\) is \(0.1230\).

round(qnorm(0.1230, lower.tail = FALSE), digits = 2)

[1] 1.16

The \(z\)-score whose area to its right is \(0.123\) is \(1.16\).

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Problem 6. Find the \(z\)-score when the area from \(z=0\) to \(z\) is \(0.3770\).

area <- 0.5-0.3770
round(qnorm(area, lower.tail = FALSE), digits = 2)

[1] 1.16

Notice that the upper z for the given interval has no preceding \(-\) sign, implying that we are to consider a \(+z\) value. Here, the \(z\) score when \(0.377\) is the area from \(z=0\) to this \(z\) value is \(1.16\).

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Problem 7.a. What \(z\)-score correspond to an area of 0.01 to the right of a positive \(z\) value?.

round(qnorm(0.01, lower.tail = FALSE), digits = 2)

[1] 2.33

The corresponding \(z\) value is \(2.33\).

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Problem 7.b. What \(z\)-score correspond to an area of 0.01 to the left of a negative \(z\) value?.

round(qnorm(0.01, lower.tail = TRUE), digits = 2)

[1] -2.33

The corresponding \(z\) value is \(-2.33\).

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Problem 7.c. What \(z\)-score correspond to an area of 0.05 to the right of a positive \(z\) value?.

round(qnorm(0.05, lower.tail = FALSE), digits = 2)

[1] 1.64

The corresponding \(z\) value is \(1.64\).

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Problem 7.d. What \(z\)-score correspond to an area of \(0.05\) to the left of a negative \(z\) value?.

round(qnorm(0.05, lower.tail = TRUE), digits = 2)

[1] -1.64

The corresponding \(z\) value is \(-1.64\).

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Problem 7.e. What \(z\)-scores correspond to an area of \(0.90\) between the positive and negative values of \(z\)?.

# Determine the area to the left of the $-Z$ value or that to the right of the $+z$ value. 
area <- (1-0.90)/2
round(qnorm(area, lower.tail = TRUE), digits = 2)

[1] -1.64

The \(z\) scores are \(-1.64\) and \(+1.64\).

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Problem 7.f. What \(z\)-scores correspond to an area of \(0.99\) between the positive and negative values of \(z\)?.

area <- (1-0.99)/2
round(qnorm(area, lower.tail = TRUE), digits = 2)

[1] -2.58

The \(z\) scores are \(-2.58\) and \(+2.58\).

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Problem 8.a. Evaluate \(P(-0.82<= z < 1.30)\).

round(pnorm(1.30)-pnorm(-0.82), digits = 4)

[1] 0.6971

The probability of a \(z\)-score greater than or equal to \(-0.82\) but less than \(1.30\) is \(0.6971\).

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Problem 8.b. Evaluate \(P(-2.60 < z <= -0.55)\).

round(pnorm(-0.55)-pnorm(-2.60), digits = 4)

[1] 0.2865

The probability of a \(z\) value greater than \(-2.60\) but lesser than or equal to \(-0.55\) is \(0.2865\).

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Problem 8.c. Evaluate \(P(1.07 <= z <= 2.81)\).

round(pnorm(2.81)-pnorm(1.07), digits = 4)

[1] 0.1398

The probability of a \(z\) value greater than or equal to \(1.07\) but lesser than or equal to \(2.81\) is \(0.1398\).