Ex a m p l e 2
The heights (in centim eters) of 50 w om en in a m ovie theater are shown below.
Use a graphing calculator to make a histogram of the data, and find the m ean and
standard deviation. Then determ ine whether the data are approxim ately norm ally
distributed.
162,170,173,163,157,168,162,153,158,177,162,159,169,161,170,160,156,
183,166,152,174,160,157,163,158,154,146,167,165,153,160,162,161,154,
165,167,170,158,159,167,166,156,161,163,158,161,161,166,163,148
Press , select Edit..., an d en ter th e d ata into list LI. Use th e gjgjjQ Q ) feature to
draw a histogram of the data, an d use th e 1-Var Stats feature to calculate th e m ean
a n d standard deviation. Recall th at th e Greek letter a represents stan d ard deviation.1- Var Stats
x = 162.08
Ix = 8104
Ix 2 = 1315912
Sx = 7 .0 2 1 3 6 6 8 0 7
ax =6.950798515
In = 50The histogram is bell-shaped an d sym m etric ab o u t th e m ean, w hich is one
characteristic of a n orm al distribution. The m ean is about 162.1 cm a n d the standard
deviation is about 7.0 cm. Find th e p ercen t of th e d ata values th a t lie w ithin one, two,
an d th ree stan d ard deviations of the m ean.
W ithin ler of m e a n (betw een 155.1 cm an d 169.1 cm): 36 values, or 72%
Within 2a of m e a n (betw een 148.1 cm an d 176.1 cm): 46 values, or 92%
Within 3a of m ean (betw een 141.1 cm an d 183.1 cm): 50 values, or 100%
The d ata com e close to following th e 68%-95%-99.7% rule for a n orm al distribution.
So th e d ata are approxim ately norm ally distributed.f ——..............— \
Ex e r c i se s
- Boxes of cereal are filled by a m achine. The weights of cereal in the boxes are
norm ally distributed w ith a m e a n of 16 oz an d a stan d ard deviation of 0.25 oz.
a. W hat p ercen t of boxes contain betw een 15.5 oz a n d 16 oz of cereal?
b. W hat p ercen t of boxes contain m ore th a n 16.25 oz of cereal?
c. W hat p ercen t of boxes contain b etw een 15.8 oz a n d 16.15 oz of cereal? - The prices of 50 TVs sold at an electronics store are show n below. Use a graphing
calculator to m ake a histogram of the data, an d find th e m e a n a n d standard
deviation. Are th e d ata approxim ately norm ally distributed? Explain.
$880, $205, $945, $230, $355, $540, $365, $600, $200, $615, $825, $860, $745, $730,
$790, $945, $205, $220, $600, $665, $835, $390, $880, $590, $460, $900, $430, $835,
$280, $920, $835, $570, $710, $265, $895, $270, $380, $565, $515, $520, $465, $425,
$555, $585, $255, $750, $725, $330, $905, $280
784 Co n cep t By t e Normal Distributions