Business Statistics I: Step-by-Step Guidance for Practice Final Questions

Q1. A study of the time spent shopping in a supermarket for a market basket of specific items showed an approximately uniform distribution between 20 minutes and 40 minutes. What is the probability that the shopping time will be less than 35 minutes?

Background

Topic: Uniform Probability Distribution

This question tests your understanding of the uniform distribution and how to calculate probabilities for intervals within a uniform distribution.

Key Terms and Formulas

• Uniform Distribution: A probability distribution where all outcomes are equally likely within a certain interval.

• Probability Density Function (PDF) for a uniform distribution from $a$ to $b$:

$f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

• Probability that $X$ is less than some value $c$ (where $a \leq c \leq b$):

$P(X < c) = \frac{c - a}{b - a}$

Step-by-Step Guidance

1. Identify the endpoints of the uniform distribution: $a = 20$ minutes, $b = 40$ minutes.

2. Determine the value for which you want the probability: $c = 35$ minutes.

3. Set up the formula for the probability: $P(X < 35) = \frac{35 - 20}{40 - 20}$

4. Calculate the numerator and denominator separately before dividing.

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Q2. The time between arrivals at an intersection follows an exponential probability distribution with a mean of 30 seconds. What is the probability the arrival time between vehicles is 32 seconds or less?

Background

Topic: Exponential Probability Distribution

This question tests your ability to use the exponential distribution to find the probability that a random variable is less than a certain value.

Key Terms and Formulas

• Exponential Distribution: Used to model the time between events in a Poisson process.

• Mean ($\mu$): The average time between arrivals.

• Rate parameter ($\lambda$): $\lambda = \frac{1}{\mu}$

• Cumulative Distribution Function (CDF):

$P(X \leq x) = 1 - e^{-\lambda x}$

Step-by-Step Guidance

1. Identify the mean: $\mu = 30$ seconds.

2. Calculate the rate parameter: $\lambda = \frac{1}{30}$

3. Set up the CDF formula for $x = 32$ seconds: $P(X \leq 32) = 1 - e^{-\lambda \times 32}$

4. Substitute $\lambda$ and $x$ into the formula and simplify the exponent.

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Q3. Given a normal distribution with $\mu=40$ and $\sigma=2$, between what two X values (symmetrically distributed around the mean) are 80 percent of the values?

Background

Topic: Normal Distribution and Symmetric Intervals

This question tests your understanding of the normal distribution and how to find symmetric intervals around the mean that contain a specified percentage of the data.

Key Terms and Formulas

• Normal Distribution: A bell-shaped, symmetric probability distribution defined by its mean ($\mu$) and standard deviation ($\sigma$).

• Z-score: $z = \frac{X - \mu}{\sigma}$

• To find the interval containing 80% of the data, find $z$ such that $P(-z < Z < z) = 0.80$

Step-by-Step Guidance

1. Recognize that you need to find $z$ such that the area between $-z$ and $z$ is 0.80.

2. Use the standard normal table (or calculator) to find the $z$ value corresponding to the middle 80% (i.e., 10% in each tail).

3. Once you have $z$, convert it back to $X$ values using $X = \mu \pm z\sigma$.

4. Set up the expressions for the lower and upper bounds: $X_{lower}=40-.84\times2$, $X_{upper} = 40 + z \times 2$.

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Q4. A study of the time spent shopping in a supermarket for a market basket of specific items showed an approximately uniform distribution between 10 minutes and 40 minutes. Compute the mean and standard deviation.

Background

Topic: Uniform Distribution - Mean and Standard Deviation

This question tests your ability to calculate the mean and standard deviation for a continuous uniform distribution.

Key Terms and Formulas

• Uniform Distribution: All outcomes between $a$ and $b$ are equally likely.

• Mean: $\mu = \frac{a + b}{2}$

• Standard Deviation: $\sigma = \frac{b - a}{\sqrt{12}}$

Step-by-Step Guidance

1. Identify the endpoints: $a = 10$, $b = 40$.

2. Calculate the mean: $\mu = \frac{10 + 40}{2}$

3. Calculate the standard deviation: $\sigma = \frac{40 - 10}{\sqrt{12}}$

4. Simplify the expressions for both mean and standard deviation.

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Q5. If a random sample of 50 people are selected, 42 of them are classified as employed. What is the sample proportion and standard error of proportion if the population proportion is 85% (0.85)?

Background

Topic: Sample Proportion and Standard Error

This question tests your understanding of how to calculate the sample proportion and the standard error of a proportion, given a sample and a known population proportion.

Key Terms and Formulas

• Sample Proportion: $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes and $n$ is the sample size.

• Standard Error of Proportion: $SE = \sqrt{\frac{p(1-p)}{n}}$

• $p$ is the population proportion.

Step-by-Step Guidance

1. Calculate the sample proportion: $\hat{p} = \frac{42}{50}$

2. Identify the population proportion: $p = 0.85$

3. Calculate the standard error: $SE = \sqrt{\frac{0.85 \times (1 - 0.85)}{50}}$

4. Simplify the numerator and denominator inside the square root before calculating the standard error.

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Q6. The median salary is $60,000 and the mean salary of the population is $65,000. a) If a sample of 9 is selected, is the data definitely normally distributed? b) If a sample of 81 is selected, is the data definitely normally distributed?

Background

Topic: Normality and the Central Limit Theorem

This question tests your understanding of the normality assumption and the Central Limit Theorem (CLT) as it applies to sample means.

Key Terms and Concepts

• Normal Distribution: A symmetric, bell-shaped distribution.

• Central Limit Theorem: For large sample sizes, the sampling distribution of the sample mean approaches normality, regardless of the population's distribution.

• Median vs. Mean: If the mean and median are not equal, the distribution is skewed.

Step-by-Step Guidance

1. Note that the mean ($65,000$) is greater than the median ($60,000$), indicating a right-skewed distribution.

2. For part (a), consider the sample size ($n=9$) and whether the CLT applies for small samples.

3. For part (b), consider the larger sample size ($n=81$) and how the CLT affects the sampling distribution of the mean.

4. Think about whether the original data or the sample mean is normally distributed in each case.

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Q7. Given a standardized normal distribution, what is the probability that Z is between -0.75 and 0.5?

Background

Topic: Standard Normal Distribution (Z-distribution)

This question tests your ability to use the standard normal table to find the probability that a Z-score falls within a given interval.

Key Terms and Formulas

• Standard Normal Distribution: A normal distribution with mean 0 and standard deviation 1.

• Z-score: Represents the number of standard deviations from the mean.

• Probability between two Z-scores: $P(a < Z < b) = P(Z < b) - P(Z < a)$

Step-by-Step Guidance

1. Look up the cumulative probability for $Z = 0.5$ in the standard normal table: $P(Z < 0.5)$.

2. Look up the cumulative probability for $Z = -0.75$: $P(Z < -0.75)$.

3. Subtract the two probabilities: $P(-0.75 < Z < 0.5) = P(Z < 0.5) - P(Z < -0.75)$

4. Set up the subtraction, but do not compute the final value yet.

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Q8. A survey showed that 65% of people prefer working from home rather than commuting to work. Suppose you select a sample of 81 workers, what is the probability that in the sample between 62% and 68% of the sample proportion prefer working from home?

Background

Topic: Sampling Distribution of the Sample Proportion

This question tests your ability to use the normal approximation to the binomial distribution for sample proportions.

Key Terms and Formulas

• Population proportion: $p = 0.65$

• Sample size: $n = 81$

• Standard error: $SE = \sqrt{\frac{p(1-p)}{n}}$

• Z-score for sample proportion: $z = \frac{\hat{p} - p}{SE}$

• Probability: $P(a < \hat{p} < b) = P(z_a < Z < z_b)$

Step-by-Step Guidance

1. Calculate the standard error: $SE = \sqrt{\frac{0.65 \times 0.35}{81}}$

2. Find the Z-score for $\hat{p} = 0.62$: $z_1 = \frac{0.62 - 0.65}{SE}$

3. Find the Z-score for $\hat{p} = 0.68$: $z_2 = \frac{0.68 - 0.65}{SE}$

4. Use the standard normal table to find the probabilities corresponding to $z_1$ and $z_2$.

5. Set up the probability: $P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$

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Q9. The diameter of a brand of tennis balls is approximately normally distributed, with a mean of 2.64 inches and a standard deviation of 0.05 inch. A random sample of 16 tennis balls is selected. Between what two values of X (symmetrically distributed around the mean) are 60% of the data?

Background

Topic: Sampling Distribution of the Sample Mean

This question tests your ability to find symmetric intervals around the mean for a sample mean, using the normal distribution.

Key Terms and Formulas

• Mean of sample mean: $\mu_{\bar{x}} = \mu$

• Standard error: $SE = \frac{\sigma}{\sqrt{n}}$

• Symmetric interval: $\mu \pm z \times SE$

• Find $z$ such that $P(-z < Z < z) = 0.60$

Step-by-Step Guidance

1. Calculate the standard error: $SE = \frac{0.05}{\sqrt{16}}$

2. Find the $z$ value such that the middle 60% of the distribution is between $-z$ and $z$ (i.e., 20% in each tail).

3. Calculate the lower and upper bounds: $X_{lower} = 2.64 - z \times SE$, $X_{upper} = 2.64 + z \times SE$

4. Set up the expressions for the bounds, but do not compute the final values yet.

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Q10a. Given a normal distribution with $\mu=80$ and $\sigma=4$, what is the probability that $X>83$?

Background

Topic: Normal Distribution - Calculating Probabilities

This question tests your ability to find the probability that a value from a normal distribution exceeds a certain value.

Key Terms and Formulas

• Z-score: $z = \frac{X - \mu}{\sigma}$

• Probability: $P(X > x) = 1 - P(X < x)$

Step-by-Step Guidance

1. Calculate the Z-score for $X = 83$: $z = \frac{83 - 80}{4}$

2. Look up $P(Z < z)$ in the standard normal table.

3. Subtract this value from 1 to get $P(X > 83)$.

4. Set up the subtraction, but do not compute the final value yet.

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Q10b. Given a normal distribution with $\mu=80$ and $\sigma=4$, five percent of the values are less than what X value?

Background

Topic: Normal Distribution - Percentiles

This question tests your ability to find the value of $X$ corresponding to a given percentile in a normal distribution.

Key Terms and Formulas

• Percentile: The value below which a given percentage of observations fall.

• Z-score for the 5th percentile: Use the standard normal table to find $z$ such that $P(Z < z) = 0.05$

• Convert Z-score to X: $X = \mu + z\sigma$

Step-by-Step Guidance

1. Find the Z-score corresponding to the 5th percentile using the standard normal table.

2. Plug the Z-score into the formula: $X = 80 + z \times 4$

3. Set up the calculation, but do not compute the final value yet.

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Q11a. The weight of a product is normally distributed with a mean of 80 grams with a standard deviation of 0.04 gram. What is the probability that an individual product contains less than 80.5 grams?

Background

Topic: Normal Distribution - Calculating Probabilities

This question tests your ability to find the probability that a value from a normal distribution is less than a given value.

Key Terms and Formulas

• Z-score: $z = \frac{X - \mu}{\sigma}$

• Probability: $P(X < x) = P(Z < z)$

Step-by-Step Guidance

1. Calculate the Z-score for $X = 80.5$: $z = \frac{80.5 - 80}{0.04}$

2. Look up $P(Z < z)$ in the standard normal table.

3. Set up the probability statement, but do not compute the final value yet.

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Q11b. The weight of a product is normally distributed with a mean of 80 grams with a standard deviation of 0.04 gram. If a sample of 16 products is selected, what is the probability that the sample mean weight is less than 80.5 grams?

Background

Topic: Sampling Distribution of the Sample Mean

This question tests your ability to use the sampling distribution of the mean to find probabilities for sample means.

Key Terms and Formulas

• Standard error: $SE = \frac{\sigma}{\sqrt{n}}$

• Z-score for sample mean: $z = \frac{\bar{x} - \mu}{SE}$

• Probability: $P(\bar{X} < x) = P(Z < z)$

Step-by-Step Guidance

1. Calculate the standard error: $SE = \frac{0.04}{\sqrt{16}}$

2. Calculate the Z-score for $\bar{x} = 80.5$: $z = \frac{80.5 - 80}{SE}$

3. Look up $P(Z < z)$ in the standard normal table.

4. Set up the probability statement, but do not compute the final value yet.

Try solving on your own before revealing the answer!