A quality control analyst measured the shelf life, in days, of a sample of 80 manufactured… — GMAT

A quality control analyst measured the shelf life, in days, of a sample of $80$ manufactured batteries, where each battery had a distinct shelf life. Battery $X$ had a shelf life at the $65\text{th}$ percentile of the sample. If $20$ additional batteries are subsequently tested and every one of these $20$ batteries has a shelf life strictly greater than Battery $X$ , what is the percentile rank of Battery $X$ 's shelf life in the combined sample of $100$ batteries?

$52\text{nd}$ percentileAnswer
B
$45\text{th}$ percentile
C
$51\text{st}$ percentile
D
$60\text{th}$ percentile
E
$65\text{th}$ percentile

Answer

The

52\text{nd}

percentile

In the initial sample of

80

batteries, a shelf life at the

65\text{th}

percentile means that

65\%

80

batteries, or

52

batteries, have a shelf life less than or equal to Battery

X

. When

20

new batteries are added—all with shelf lives strictly greater than Battery

X

—the new total number of batteries is

100

, while the number of batteries with shelf life less than or equal to Battery

X

remains

52

. Thus, Battery

X

is at the

\frac{52}{100} \times 100\% = 52\text{nd}

percentile of the combined sample.

Step-by-Step Solution

Determine the number of batteries in the original sample with a shelf life less than or equal to Battery X

Since

65\%

of the original

80

batteries are at or below Battery

X

0.65 \times 80 = 52

batteries.

By definition, a percentile rank of

65

in a dataset of size

N

represents

65\%

of the data points at or below that score.

Calculate the total number of batteries in the combined dataset

80 + 20 = 100

batteries.

Adding

20

new batteries increases the sample size from

80

100

Determine the number of batteries at or below Battery X in the combined dataset

The count remains

52

batteries.

All

20

newly added batteries have shelf lives strictly greater than Battery

X

, so none of them fall at or below Battery

X

Calculate the new percentile rank of Battery X

\frac{52}{100} \times 100\% = 52\text{nd percentile}.

The percentile rank is the percentage of total data points in the combined sample that are less than or equal to Battery

X

Key Concept

Percentile rank represents the proportion of values in a dataset that are less than or equal to a given value. When a dataset expands by adding values strictly above a target value, the number of values at or below the target remains constant while the total sample size increases.

Estimated Time:1m 30s

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