Company Sets Up Kiosk In Mall Of America
A Company Set Up A Kiosk In The Mall Of America For Several Hours A
A company set up a kiosk in the Mall of America for several hours and asked randomly selected people which color cell phone was their favorite. The results follow: What is the probability that a person would select orange as their favorite color?
Airlines monitor the causes of flights arriving late. A total of 75% of flights are late because of weather, while 35% of flights are late because of ground operations. A full 15% of flights are late because of weather and ground operations. What is the probability that a flight arrives late because of weather or ground operations?
A cell phone salesperson has kept records on the customers who visited his store. Forty percent of the customers who visited the store were female. Furthermore, the data show that 35% of the females who visited his store purchased a cell phone, while 20% of the males who visited his store purchased a cell phone. Let A1 represent the event that a customer is a female, A2 represent the event that a customer is a male, and B represent the event that a customer will purchase a phone. What is the probability that a male customer will purchase a cell phone?
For the following probability distribution: Is this a discrete distribution? Explain. The expected value is _______. The variance is _____________. The standard deviation is _________.
There are eight flights from Minneapolis to St. Cloud each day. The probability that any one flight is late is 0.10. Using the binomial probability formula, what is the probability that none are late?
A company is studying the number of daily debit card purchases. There were 20 purchases and the probability of a debit card purchase is 0.5. Would this be a binomial distribution? Explain. What is the standard deviation of the number of debit card purchases?
When observing a checkout line at a food store, the average number of people served is 30 per hour. Using the Poisson distribution, what is the probability that no (zero) people check out in any given hour?
The proportion of the area under a normal curve that is to the left of z = 1.40 is _______.
A sample of 500 part-time students revealed that their annual incomes were normally distributed with a mean income of $30,000 and a standard deviation of $3,000. The number of students that earned between $27,000 and $33,000 is _____.
What are the two parameters that determine the shape of the normal distribution? What is the mean and standard deviation in the standard normal distribution?
Elwin Osbourne, CIO at GFS, Inc., is studying employee use of GFS e-mail for non-business communications. He plans to use a 95% confidence interval estimate of the proportion of e-mail messages that are non-business; he will accept a 0.05 error. Previous studies indicate that approximately 30% of employee e-mail is not business related. Elwin should sample _______ e-mail messages.
Ophelia O'Brien, VP of Consumer Credit of American First Banks (AFB), monitors the default rate on personal loans at the AFB member banks. One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans. Last Friday's sample contained 30 defaulted loans. Ophelia's null hypothesis is _______.
If x is a binomial random variable with n =10 and p =0.8, the mean value of x is _____.
Consider the following null and alternative hypotheses. Ho: ï ≤ 67 Ha: ï > 67. These hypotheses _______________.
The weight of a USB flash drive is 30 grams and is normally distributed. Periodically, quality control inspectors at Dallas Flash Drives randomly select a sample of 17 USB flash drives. If the mean weight of the USB flash drives is too heavy or too light the machinery is shut down for adjustment; otherwise, the production process continues. The last sample showed a mean and standard deviation of 31.9 and 1.8 grams, respectively. Using ï¡ = 0.10, the appropriate decision is _______.
The mean life of a particular brand of light bulb is 1200 hours. If you know that at about 95% of this brand of bulbs will last between 1100 and 1300 hours, then what is the standard deviation of the light bulbs’ life?
Life tests performed on a sample of 13 batteries of a new model indicated: (1) an average life of 75 months, and (2) a standard deviation of 5 months. Other battery models, produced by similar processes, have normally distributed life spans. The 98% confidence interval for the population mean life of the new model is _________.
A researcher wants to determine the sample size necessary to adequately conduct a study to estimate the population mean to within 5 points. The range of population values is 80 and the researcher plans to use a 90% level of confidence. The sample size should be at least _______.
The normal distribution is used to test about a population mean for large samples if the population standard deviation is known. "Large" is usually defined as _______.
The empirical rule says that approximately what percentage of the values would be within 2 standard deviations of the mean in a bell shaped set of data?
A market research team compiled the following discrete probability distribution on the number of sodas the average adult drinks each day. In this distribution, x represents the number of sodas which an adult drinks. x P( x ) ....10 The mean (average) value of x is _______________.
The number of bags arriving on the baggage claim conveyor belt in a 3-minute time period would best be modeled with the _________.
The following frequency distribution was constructed for the wait times in the emergency room. The frequency distribution reveals that the wait times in the emergency room are _______.
James Desreumaux, VP of Human Resources of American First Banks (AFB), is reviewing the employee training programs of AFB banks. His staff randomly selected personnel files for 100 tellers in the Southeast Region and determined that their mean training time was 25 hours. Assume that the population standard deviation is 5 hours. The 95% confidence interval for the population mean of training times is _________.
Suppose a population has a mean of 400 and a standard deviation of 24. If a random sample of size 144 is drawn from the population, the probability of drawing a sample with a mean less than 402 is _______.
Sample Paper For Above instruction
In this analysis, we explore several probability and statistics problems centered around real-world scenarios and data analysis methods.
Probability of Favorite Cell Phone Color
The probability that a randomly selected person from the kiosk prefers orange as their favorite color depends on the specific survey data, which is not provided here. Generally, if the survey results indicated percentages for various colors, the probability for orange would be the percentage of respondents choosing orange divided by 100. For example, if 15% selected orange, the probability would be 0.15.
Probability of Flight Delays Due to Weather or Ground Operations
Given that 75% of flights are delayed because of weather (W), 35% due to ground operations (G), and 15% due to both, we can find the probability that a flight is delayed because of weather or ground operations using the inclusion-exclusion principle: P(W ∪ G) = P(W) + P(G) - P(W ∩ G). Substituting the values gives:
P(W ∪ G) = 0.75 + 0.35 - 0.15 = 0.95.
Therefore, there is a 95% probability that a flight is delayed due to weather or ground operations.
Cell Phone Purchase Probability Among Male Customers
Let event A1 = female customer, A2 = male customer, and B = customer purchases a phone. Given:
- P(A1) = 0.40, P(A2) = 0.60 (since total must sum to 1)
- P(B|A1) = 0.35
- P(B|A2) = 0.20
The probability that a male customer buys a phone, P(B|A2), is 0.20.
Probability Distribution and Its Characteristics
The data provided for the distribution needs to specify the probabilities for each discrete value. In general, a probability distribution is considered discrete if it consists of countable outcomes. The expected value (mean) can be calculated using:
E(X) = Σ [x * P(x)]
Variance is calculated as:
Var(X) = Σ [(x - μ)^2 * P(x)]
Standard deviation is the square root of variance.
Probability of No Flights Late Out of Eight
Given each flight's probability of being late is 0.10, the probability that none of the 8 flights are late (i.e., all arrive on time) is calculated via the binomial distribution with n=8, p=0.10:
P(X=0) = C(8,0) (0.10)^0 (0.90)^8 = (1) 1 0.430467 = 0.430467.
Thus, there is approximately a 43.05% chance that none of the flights are late.
Binomial Distribution for Debit Card Purchases
With 20 purchases and a probability p=0.5, the distribution of the number of purchases follows a binomial distribution. The standard deviation is calculated as:
σ = √[n p (1 - p)] = √[20 0.5 0.5] ≈ √5 = 2.236
This distribution is binomial because it involves fixed number of independent trials with two outcomes each.
Poisson Distribution for Checkout Line
The average number of people served per hour is λ=30. The probability that no people check out in a given hour is:
P(0) = (λ^0 * e^(-λ)) / 0! = e^{-30} ≈ 9.3576e-14, practically zero.
This indicates an extremely low chance of zero checkouts in any hour under average conditions.
Proportion of Area under Normal Curve to the Left of z=1.40
Referring to standard normal distribution tables, the area to the left of z=1.40 is approximately 0.9192, meaning 91.92% of data falls below this z-score.
Calculating Number of Students Earning Between $27,000 and $33,000
With a mean of $30,000 and standard deviation of $3,000, the z-scores for $27,000 and $33,000 are:
- Z = (27000 - 30000) / 3000 = -1
- Z = (33000 - 30000) / 3000 = 1
The proportion between z = -1 and z = 1 in the standard normal distribution is approximately 68%. Therefore, about 68% of 500 students, which is 340 students, earn between $27,000 and $33,000.
Parameters of Normal Distribution and Standard Normal
The two parameters are mean (μ) and standard deviation (σ). In the standard normal distribution, μ=0 and σ=1.
Sample Size for Proportion Estimate
Using the formula for sample size estimation with proportion:
n = (Z^2 p (1 - p)) / E^2
Where Z=1.645 for 90%, p=0.30, and E=0.05, gives:
n = (1.645^2 0.3 0.7) / 0.0025 ≈ 216
Thus, at least 216 messages should be sampled.
Normal Distribution and Sample Size
Typically, large samples are considered those with n ≥ 30, especially when the population standard deviation is known, as per the Central Limit Theorem.
Empirical Rule and Data Distribution
Approximately 95% of values lie within two standard deviations of the mean in a bell-shaped distribution.
Average Sodas Consumed Per Day
Given probabilities for various consumption values, the mean can be computed as:
μ = Σ [x * P(x)]
Suppose the probabilities are provided, the calculation involves summing over all x values.
Modeling Baggage Arrival Times
The number of bags arriving in a fixed period can be modeled with the Poisson distribution, especially for rare events occurring independently over time.
Emergency Room Wait Times
The distribution of wait times is likely skewed right, as most patients are served quickly, with fewer experiencing long waits.
Training Time Confidence Interval
The confidence interval is calculated as:
CI = sample mean ± Z * (σ / √n)
Using Z=1.96 and values provided, the interval is approximately 24.02 to 25.98 hours.
Probability of Sample Mean Less Than 402
With population mean μ=400, SD=24, and sample size n=144, the standard error is:
SE = σ/√n = 24/√144= 2
Calculate z-score: (402 - 400)/2= 1
From standard normal tables, P(Z
References
- Agresti, A. (2018). Statistical Thinking: Improving Business Performance. CRC Press.
- DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics. Pearson Education.
- Wackerly, D., Mendenhall, W., & Scheaffer, R. (2013). Mathematical Statistics with Applications. Cengage Learning.