Verified answer

You are correct at the start 6.3% = 0.063

No look at the Z-score table for normal distribution.

The corresponding z-score from row and column headings

0.063 corresponds to -1.5 and 0.03. It is -1.5 + 0.03 = -1.53

Z-score is -1.53

The links show z=-1.53

https://gyazo.com/bcbeec04886f110c957d3c907266ce6b

https://gyazo.com/12d58bd9335a386b7280e21792389699

If you sketch a bell curve and put a vertical line where only a small fraction of the area lies to its left, you are putting the line way to the left of center, so the z score is negative.

You want 1.000 - 0.063 = 0.937 to appear in the FACE of the table.

You will see that z = 1.53 corresponds to an area of 0.9370,

so no interpolation is needed.

The z score that has 6.3 percent of the distribution to its left is -1.53.

If you use a table that goes only from area = 0.0000 to area = 0.4999,

you'd be looking for 0.437 instead of 0.937.

We can construct the table in R:

# A list of z-scores from -4 to 4 in increments of + 0.001

> z_scores = seq(from = -4, to = 4, by = 0.001)

# Probabilities for each of the z-scores.

> probabilities = pnorm(q = z_scores, lower.tail = TRUE)

# Round the probabilities to 3 decimal places:

> probabilities = round(probabilities, 3)

# Put the z-scores and the probabilities into a dataframe:

> frame = data.frame(z_scores, probabilities)

# Subset the z-scores which have a probability between 0.062 and 0.064 since there isn't one that is exactly 0.063

> frame$z_scores[frame$probabilities == 0.063]

[1] -1.534 -1.533 -1.532 -1.531 -1.530 -1.529 -1.528

[8] -1.527

So any of the z-scores from -1.527 to -1.534 has a probability of approximately 0.063 precise to at least 3 decimal places.

If you plot the Area under the curve against the z-scores you see that the z-score corresponds to about 6.3% is about -1.53

0.063 is the value you are looking for inside the table.

It depends on what type of table you have. Since the bell curve is symmetric, if then they just give you the right hand part.

So, you are equivalently looking for a z such that the value in the table is 0.063. This is for tables that show you whats to the right of your z; that is from z to ∞.

Some tables give whats from z=0 to z = z. If thats the kind of table you have, then the half bell curve has an area of 0.5.

To the right of your sought z its 0.063. So to its left its 0.5-0.063 = 0.437. So look in your table for a z such that in the table its 0.437.

But dont forget, you really want the left side. So your answer would be - z.