6.2 Using the Normal Distribution with Technology- Calculator

Suppose that the weight of sweet cherries is normally distributed with mean μ=6 ounces and standard deviation σ=1.4 ounces. What proportion of sweet cherries weigh less than 5 ounces?

0.2375 Let X represent the random variable. The mean is μ=6, and the standard deviation is σ=1.4. The lower bound is negative infinity, which is represented by −9999, and the upper bound is 5. Now use a TI-83, TI-83 plus, or TI-84 calculator to find the probability. 1. Press 2nd and VARS for the DISTR menu. 2. Press 2 for normalcdf(. 3. Enter −9999,5,6,1.4) and press ENTER. The answer, rounded to four decimal places, is P(X<5)≈0.2375.

Dakota is a quality control manager for a manufacturing plant that produces batteries for tablet computers. Based on prior history, Dakota knows that 0.6% of all tablet computer batteries produced are defective. He selects a random sample of n=1,200 tablet computer batteries from a recently produced lot and tests each battery to determine whether it is defective. What is the probability that Dakota finds 5 or fewer defective batteries? Use a TI-83, TI-83 plus, or TI-84 calculator to find the probability.

0.2626

The probability of an airline flight arriving on time at a certain airport is 84%. Use the normal approximation to find the probability that more than 340 in a random sample of 400 commercial airline flights at the airport will arrive on time. Use a TI-83, TI-83 plus, or TI-84 calculator to find the probability.

0.2697

A bank found that 2.5% of its loans to new small businesses become delinquent. If 400 small businesses are selected randomly from the bank's files, what is the probability that at least 12 of them are delinquent? Use a TI-83, TI-83 plus, or TI-84 calculator to find the probability. Round your answer to four decimal places.

0.3155

Suppose that the annual household income in a region is normally distributed with mean $25,000 and standard deviation $6,000. If the poverty level for the region is at or below $12,000, what percentage of the population lives in poverty? Round your answer to two decimal places.

1.51%

In a certain town, 10% of the population is left handed. If 85 residents of the town are selected at random, what is the probability that 3 or fewer will be left handed? Use a TI-83, TI-83 plus, or TI-84 calculator to find the probability. Round your answer to four decimal places.

0.0353 Here, n=85 and p=10%=0.1. So, np=85×0.1=8.5 and n(1−p)=85×(1−0.1)=85×0.9=76.5 are both greater than 5, which means that the normal distribution can be used to approximate the binomial. As the probability of at most 3 has to be calculated, the value of interest is 3+0.5=3.5, as the probability does include 3. Use a TI-83, TI-83 plus, or TI-84 calculator to calculate the probability under the normal curve to the left of the value of interest. 1. Press 2nd and VARS for the DISTR menu. Press 2 for normalcdf(. 2. Enter −9999,3.5,85×0.1,√(85×0.1×0.9)) and press ENTER. The answer, rounded to four decimal places, is 0.0353. Thus, the probability that 3 or fewer residents out of 85 will be left handed is approximately 0.0353.

The probability of someone buying a particular newspaper in a town is 0.4. Out of 80 customers who come into a shop, what is the probability of at most 26 customers buy this newspaper? Use the normal distribution to approximate the binomial distribution.

0.1047

The diameters of apples from a certain farm follow normal distribution with mean 4 inches and standard deviation 0.4 inch. Apples can be size-sorted by being made to roll over mesh screens. First the apples are rolled over a screen with mesh size 3.5 inches. This separates out all the apples with diameters less than 3.5 inches. Second, the remaining apples are rolled over a screen with mesh size 4.3 inches. Find the proportion of apples with diameters greater than 4.3 inches.

0.2266 Let X represent the random variable. The mean is μ=4, and the standard deviation is σ=0.4. The lower bound is 4.3, and the upper bound is positive infinity, which is represented by 9999. Now use a TI-83, TI-83 plus, or TI-84 calculator to find the probability. 1. Press 2nd and VARS for the DISTR menu. 2. Press 2 for normalcdf(. 3. Enter 4.3,9999,4,0.4) and press ENTER. The answer, rounded to four decimal places, is P(X>4.3)≈0.2266.

A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myriad of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour. Find the probability that a household personal computer is used for entertainment between 2.1 and 2.45 hours per day. Round your answer to four decimal places.

0.2367

The burning times of scented candles, in minutes, are normally distributed with a mean of 249 and a standard deviation of 20. Find the number of minutes a scented candle burns if it burns for a shorter time than 60% of all scented candles.

243.93 the mean, μ, is 249 standard deviation, σ, is 20. In this case, 60% of scented candles burn longer than this scented candle, so the area to the right of x is 60%=0.60. So the area to the left of x is 1−0.60=0.40. The value of x can be calculated using a TI-83, TI-83 plus, or TI-84 calculator. 1. Press 2nd and then VARS for the DISTR menu. 2. Press 3 for invNorm(. 3. Enter 0.40,249,20) and press ENTER. The answer, rounded to two decimal places, is x≈243.93. Thus, a scented candle that burns for a shorter time than 60% of all scented candles burns for 243.93 minutes.

A credit card company receives numerous phone calls throughout the day from customers reporting fraud and billing disputes. Most of these callers are put "on hold" until a company operator is free to help them. The company has determined that the length of time a caller is on hold is normally distributed with a mean of 2.5 minutes and a standard deviation of 0.5 minutes. If 2.5% of the callers are put on hold for longer than x minutes, what is the value of x? Use a TI-83, TI-83 plus, or TI-84 calculator. Round your answer to two decimal places.

3.48

Assume that the salaries of elementary school teachers in a particular country are normally distributed with a mean of $38,000 and a standard deviation of $4,000. What is the cutoff salary for teachers in the top 10%?

43126 mean(μ)=38,000 standard deviation(σ) =4,000. Let x be the cutoff salary for teachers in the top 10%. The area to the right of x is 10%=0.10. So, the area to the left of x is 1−0.10=0.90. The value of x can be calculated using a TI-83, TI-83 plus, or TI-84 calculator. 1. Press 2nd and VARS for the DISTR menu. 2. Press 3 for invNorm(. 3. Enter 0.90,38000,4000) and press ENTER. The answer, rounded to the nearest dollar, is x≈43,126. Thus, the cutoff salary for teachers in the top 10% is $43,126.

A new tax law is expected to benefit "middle income" families, those with incomes between $25,000 and $35,000. If family income is normally distributed with mean $25,000 and standard deviation $6,000, what percentage of the population will benefit from the law? Round your answer to two decimal places.

Let X represent the random variable. The mean is μ=25,000, and the standard deviation is σ=6,000. The lower bound is 25,000, and the upper bound is 35,000. Now use a TI-83, TI-83 plus, or TI-84 calculator to find the probability. 1. Press 2nd and VARS for the DISTR menu. 2. Press 2 for normalcdf(. 3. Enter 25000,35000,25000,6000) and press ENTER. Then multiply the result by 100 to find a percentage. The answer, rounded to two decimal places, is 45.22%.