Statistics Cheatsheet 2



Statistics Cheatsheet

by Lei Bao

Normal Distribution

Terms and Definitions

• Normal Distribution

• Z table

• Z-score

Typical Problems

1. Assume that the random variable X is normally distributed, with mean $\mu$ = 45 and standard deviation $\sigma$ = 14. Compute the probability P (55 $<$ X $\le$ 70).

   • Standardizing a Normal Random Variable

   • $z = \frac{x - \mu}{\sigma}$

   • $z_{upper} = \frac{70 - 45}{14} = 1.79$

   • $z_{lower} = \frac{55 - 45}{14} = 0.71$

   • Z table

   • Area in left tail for $z_{upper}$: 0.9633

   • Area in left tail for $z_{lower}$: 0.7611

   • Area between $z_{upper}$ and $z_{lower}$: 0.9633 - 0.7611 = 0.2022

2. The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with mean 1252 and standard deviation 129 chips.

• What is the probability that a randomly selected bag contains between 1000 and 1400 chocolate chips?

  • $z_{upper} = \frac{1400 - 1252}{129} = 1.15$

  • $z_{lower} = \frac{1000 - 1252}{129} = - 1.95$

  • Area in left tail for $z_{upper}$: 0.8749

  • Area in left tail for $z_{lower}$: 0.0256

  • Area between $z_{upper}$ and $z_{lower}$: 0.8749 - 0.0256 = 0.8493

• What is the probability that a randomly selected bag contains fewer than 1000 chocolate chips?

  • $z = \frac{1000 - 1252}{129} = -1.95$

  • Area in left tail for $z$: 0.0256

• What proportion of bags contains more than 1200 chocolate chips?

  • $z = \frac{1200 - 1252}{129} = - 0.40$

  • Area in left tail for $z$: 0.3446

  • Area in right tail for $z$: 1 - 0.3446 = 0.6554

• What is the percentile rank of a bag that contains 1050 chocolate chips?

  • $z = \frac{1050 - 1252}{129} = - 1.57$

  • Area in left tail for $z$: 0.0582 $\simeq$ 6% (6th percentile)

3. Suppose a brewery has a filing machine that fills 12-ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.15 ounces and a standard deviation of 0.04 ounce. The company is interested in reducing the amount of extra beer that is poured into the 12 ounce bottles. The company is seeking to identify the highest 1.5% of the fill amounts poured by this machine. For what fill amount are they searching? Round to the nearest thousandth.

   • Area in the right tail: 0.015

   • Area in the left tail: 1 - 0.015 = 0.985

   • Z-score for 0.985: 2.17

   • Finding the Score

   • $x = \mu + z \sigma$

   • x = 12.15 + 2.17 $\times$ 0.04 = 12.2368 $\simeq$ 12.237