[Math] Find The Z-score with a percentile

statistics

Relatively basic question here ..

A Company wants to give gifts to the top 5% of customers. If the mean spend per customer is $ 135 – with a standard deviation of 55 – what balance should the company specify?

However, at the end of the first month it was found that 8% of customers qualified for the free gift. What has happened? Assuming that the standard deviation hasn’t changed, calculate the new mean spend per customer.

HOW do I calculate the Z Score for these questions ?

The first part of the question's z score = 1.645 which makes sense if looking at the z-table.

But according to the answer for the second part of the question the z score = 1.405. This I do not understand — the closest percent on the z-table is 0.9207 with a z-score of 1.401 — How do they calculate this ? and why is the closest percentile to 92% on the z-table not used ?

Best Answer

You may be misreading the table. If it looks like the one below (from here), the closest value to $0.92$ is $0.9207$, which is at the intersection of the row for $z=1.4$ and the column $0.01$. This would correspond to a $z$-score of $1.4+0.01=1.41$, not $1.401$.

However, $0.92$ is nearly exactly between the values $0.9192$ ($z=1.40$) and $0.9207$ ($z=1.41$) in the table, so the $z$-score for $0.92$ would be roughly the average of $1.40$ and $1.41$, or $1.405$. The exact $z$-score above which $8\%$ of the normal distribution lies is (rounded to $6$ decimal places) $1.404071$ (see this calculator, for example)

enter image description here

Related Solutions

[Math] Calculate Percentile of Skewed Dataset

The straightforward way is to use the definition of percentile (which differs a bit from text to text and software to software) and count observations. This works for data from any distribution. (Differences in definitions do not matter much for large samples.) Roughly speaking, the 90th percentile is a value below which one finds not more than 90% of the observations, and above which one finds not more than 10% of them.

Here are a few examples of percentiles for a couple of datasets in R, one normal (symmetrical) and one exponential (right-skewed). Notice that percentiles of small samples do not necessarily match percentiles of the populations from which they are sampled. (The method you have been using for normal data seems to conflate the two kinds of percentiles.) In the data displays below the numbers in brackets give the index of the first observation in the that row.

x = round(sort(rnorm(50, 100, 15)), 1);  x  # generate 50 obs from Norm mean=100, SD=15
x
 [1]  61.1  69.4  71.1  73.0  73.9  77.5  78.0  78.0  79.0  81.5
[11]  83.4  85.9  86.5  87.8  87.9  88.0  88.8  90.0  90.7  91.3
[21]  92.0  93.0  95.3  97.9  97.9  99.2  99.2 100.0 100.2 101.0
[31] 102.9 103.4 104.3 104.6 105.4 107.2 108.5 108.6 109.5 109.6
[41] 111.3 111.5 111.9 118.0 119.5 119.6 119.6 119.9 121.5 128.4
quantile(x, .9)  # 90th percentile
   90% 
119.51 
quantile(x, .7)  # 70th percentile
   70% 
105.94 
qnorm(c(.9, .7), 100, 15)  # 90th and 70th percentiles of POPULATION
[1] 119.2233 107.8660

x = round(sort(rexp(60, rate=1/50)), 1);  x  # generate 60 obs from EXP mean=50
x
 [1]   0.0   0.4   0.5   0.9   0.9   1.3   2.4   3.8   4.4   6.1
[11]   7.5   7.9   8.0   9.0   9.9  11.0  11.5  13.3  15.4  16.4
[21]  19.6  20.3  25.1  25.4  28.0  28.8  29.3  29.5  31.1  32.0
[31]  32.1  34.2  37.2  40.6  42.0  42.0  49.5  52.2  55.6  56.8
[41]  59.6  64.3  73.9  74.7  78.7  87.9  90.1  95.4  97.2 105.2
[51] 110.3 113.8 114.6 172.5 187.0 188.2 188.8 207.9 259.7 265.3
quantile(x, .9)
   90% 
173.95 
quantile(x, .7)
  70% 
67.18 
qexp(c(.9, .7), rate=1/50)  # 90th and 70th percentiles of POPULATION
115.12925  60.19864

[Math] If X is normal, P(X<0) = 0.5, and P(X>= 1) = 0.05, find P(|X|<= 0.25)

Hint: $P(|X|\leq 0.25) = P(X \leq 0.25) - P(X \leq -0.25)$

Best Answer

Related Solutions

[Math] Calculate Percentile of Skewed Dataset

[Math] If X is normal, P(X<0) = 0.5, and P(X>= 1) = 0.05, find P(|X|<= 0.25)

Related Question