Question 1

The manager of a grocery store has taken a random sample of 144 customers. The average length of time it took these 144 customers to check out was 3 minutes. It is known that the standard deviation of the population of checkout times is 1 minute. The standard error of the mean equals

Accepted Answer

The formula for standard error of the mean is standard deviation divided by the square root of the sample size. Therefore, the standard error of the mean in this case is 1 / sqrt(144) = 0.083.

Question 2

A random sample of 64 SAT scores of students applying for merit scholarships showed an average of 1400 with a standard deviation of 240. If we want to provide a 95% confidence interval for the population mean SAT score, the degrees of freedom for reading the t value is

Accepted Answer

The degrees of freedom for a t-distribution, when estimating a population mean from a sample, is calculated as the sample size minus one. Here, with a sample size of 64, the degrees of freedom is 64 - 1 = 63.

Question 3

In order to estimate the average electric usage per month, a sample of 64 houses was selected and the electric usage was determined. Assume a population standard deviation of 320 kilowatt-hours. If the sample mean is 1858 kWh, the 95% confidence interval estimate of the population mean is _____ kWh.

Accepted Answer

The 95% confidence interval for the population mean is calculated using the formula: $\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $Z$ is the Z-score corresponding to the confidence level (1.96 for 95%), $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values: $1858 \pm 1.96 \frac{320}{\sqrt{64}} = 1858 \pm 1.96 \times 40 = 1858 \pm 78.4$, which gives the interval $1779.6$ to $1936.4$.

Question 4

A sample of 100 information systems managers had an average hourly income of $40.00 with a standard deviation of $8.00. If we want to determine a 95% confidence interval for the average hourly income of the population, the value of t is

Accepted Answer

For a 95% confidence interval with a sample size of 100, we use the t-distribution. With 99 degrees of freedom (n-1), the t-value is approximately 1.993.

Question 5

The manager of a grocery store has taken a random sample of 144 customers. The average length of time it took these 144 customers to check out was 3 minutes. It is known that the standard deviation of the population of checkout times is 1 minute. With a .95 probability, the sample mean will provide a margin of error of

Accepted Answer

First, we need to use the formula for the margin of error:

Margin of Error = z*(standard deviation/square root of sample size)

Since we want to be 95% confident, we need to use the z-score that corresponds to a 95% confidence level. This is 1.96.

Therefore, the margin of error is:

Margin of Error = 1.96*(1/square root of 144) = 0.163

So, the answer is (C) 0.163.

Question 6

From a population that is not normally distributed and whose standard deviation is not known, a sample of 50 items is selected to develop an interval estimate for µ. Which of the following statements is true?​

Accepted Answer

When estimating a population mean from a sample where the population standard deviation is unknown, the t distribution is used. The degrees of freedom for the t distribution is calculated as the sample size minus one, which in this case is 50 - 1 = 49.

Question 7

In order to estimate the average electric usage per month, a sample of 64houses was selected and the electric usage was determined. Assume a population standard deviation of 320 kilowatt-hours. At 95% confidence, the size of the margin of error is

Accepted Answer

The formula to calculate the margin of error is:

Margin of Error = Z* * (standard deviation / sqrt(sample size))

Where Z* is the critical value of the standard normal distribution at the desired confidence level.

At 95% confidence level, Z* = 1.96.

Plugging in the given values:

Margin of Error = 1.96 * (320 / sqrt(64))
Margin of Error = 78.4

Therefore, the size of the margin of error is 78.4 kilowatt-hours, which is option C.

Question 8

A random sample of 25 employees of a local company has been taken. A 95% confidence interval estimate for the mean systolic blood pressure for all employees of the company is 123 to 139. Which of the following statements is valid?

Accepted Answer

The 95% confidence interval means that if we were to repeat the sampling procedure and construct a confidence interval each time, we would expect 95% of those intervals to contain the population mean systolic blood pressure. This statement describes the concept of "confidence" in a confidence interval. Options A and C are incorrect because they incorrectly interpret the confidence interval as applying to the sample or the population as a whole, respectively. Option D is incorrect because it refers to the specific sample mean, whereas the confidence interval pertains to the population mean.

Question 9

We can use the normal distribution to make confidence interval estimates for the population proportion, p, when​

Accepted Answer

The normal distribution can be used to make confidence interval estimates for the population proportion, p, when both np > 5 and n(1-p) > 5. This is known as the "success-failure" condition, which ensures that the sampling distribution of the sample proportion is approximately normal. If either np or n(1-p) is less than or equal to 5, then the distribution may not be normal and other methods for making confidence intervals should be used.

Question 10

In order to estimate the average time spent per student on the computer terminals at a local university, data were collected for a sample of 64 business students over a one-week period. Assume the population standard deviation is 1.6 hours. The standard error of the mean is

Accepted Answer

The standard error of the mean is calculated as the population standard deviation divided by the square root of the sample size. Here, it is 1.6 / sqrt(64) = 1.6 / 8 = 0.2.

Question 11

A sample of 100 information systems managers had an average hourly income of $40.00 with a standard deviation of $8.00. The value of the margin of error at 95% confidence is

Accepted Answer

We can use the formula: Margin of error = Critical value x Standard error
First, we need to find the critical value using a z-table or calculator. At 95% confidence level, the critical value is 1.96.
Next, we need to find the standard error using the formula: Standard error = Standard deviation / Square root of sample size
Standard error = 8 / square root of 100 = 0.8
Now we can plug in the values to find the margin of error:
Margin of error = 1.96 x 0.8 = 1.568
Therefore, the answer is D: 1.568.

Question 12

A sample of 100 information systems managers had an average hourly income of $40.00 with a standard deviation of $8.00. The standard error of the mean is

Accepted Answer

The standard error of the mean formula is standard deviation divided by the square root of the sample size. Therefore, the standard error of the mean in this case would be $8.00 divided by the square root of 100 (sample size), which equals $0.80 or 0.8. However, since the answer choices only show whole numbers, the closest option is C, which is 8.

Question 13

A random sample of 81 SAT scores of students applying for merit scholarships showed an average of 1400 with a standard deviation of 300. The t value needed to develop the 95% confidence interval for the population mean SAT score is

Accepted Answer

For a 95% confidence interval with a sample size greater than 30 (in this case, 81), the t-distribution should be used. The degrees of freedom (df) are n - 1 = 80. Looking up the t-value for 80 degrees of freedom for a 95% confidence interval in a t-table or using statistical software typically yields a value very close to 1.98 or 2. However, since the exact value for 80 degrees of freedom is not provided, the closest option given is 1.998, which is a reasonable approximation for the t-value in this context.

Question 14

A random sample of 64 SAT scores of students applying for merit scholarships showed an average of 1400 with a standard deviation of 240. The margin of error at 95% confidence is

Accepted Answer

The margin of error (ME) at 95% confidence for a sample mean can be calculated using the formula ME = z * (σ/√n), where z is the z-score corresponding to the confidence level (for 95% confidence, z ≈ 1.96), σ is the population standard deviation, and n is the sample size. Here, σ = 240, n = 64. So, ME = 1.96 * (240/√64) = 1.96 * 30 = 58.8, which rounds to 59.94, making option D the closest choice.

Question 15

In order to estimate the average electric usage per month, a sample of 64 houses was selected and the electric usage was determined. Assume a population standard deviation of 320 kilowatt-hours. The standard error of the mean is

Accepted Answer

The formula for standard error of the mean is:
Standard Error = (Population Standard Deviation) / sqrt(sample size)
Plugging in the values given,
Standard Error = 320 / sqrt(64) = 40. Therefore, the answer is D.

Question 16

In order to estimate the average time spent per student on the computer terminals at a local university, data were collected for a sample of 64 business students over a one-week period. Assume the population standard deviation is 1.6 hours. With a .95 probability, the margin of error is approximately

Accepted Answer

The margin of error (E) can be calculated using the formula E = Z * (σ/√n), where Z is the Z-score corresponding to the confidence level (1.96 for 95%), σ is the population standard deviation, and n is the sample size. Plugging in the given values: E = 1.96 * (1.6/√64) = 1.96 * 0.2 = 0.392.

Question 17

The manager of a grocery store has taken a random sample of 144 customers. The average length of time it took these 144 customers to check out was 3 minutes. It is known that the standard deviation of the population of checkout times is 1 minute. The 95% confidence interval for the true average checkout time (in minutes) is

Accepted Answer

To find the 95% confidence interval for the true average checkout time (μ), we can use the formula:

CI = x̄ ± z* (σ/√n)

where:

x̄ = sample mean = 3 minutes
z* = the z-value for the 95% confidence level, which is found using the Z-table or calculator and is approximately 1.96
σ = population standard deviation = 1 minute
n = sample size = 144

Plugging in the values, we get:

CI = 3 ± 1.96 * (1/√144)
CI = 3 ± 0.163
CI = (2.837, 3.163)

Therefore, the 95% confidence interval for the true average checkout time is (2.837, 3.163).

The correct answer is D.

Question 18

A random sample of 64 SAT scores of students applying for merit scholarships showed an average of 1400 with a standard deviation of 240. The 95% confidence interval for the population mean SAT score is

Accepted Answer

The correct answer is A because the 95% confidence interval for the population mean SAT score is 1340.06 to 1459.94. This interval is calculated using the formula: sample mean +/- 1.96 * (standard deviation / square root of sample size). In this case, the sample mean is 1400, the standard deviation is 240, and the sample size is 64. Plugging these values into the formula, we get: 1400 +/- 1.96 * (240 / square root of 64) = 1400 +/- 1.96 * (240 / 8) = 1400 +/- 1.96 * 30 = 1400 +/- 59.94 = 1340.06 to 1459.94.

Question 19

In order to estimate the average time spent per student on the computer terminals at a local university, data were collected for a sample of 64 business students over a one-week period. Assume the population standard deviation is 1.6 hours. If the sample mean is 9 hours, then the 95% confidence interval is _____ hours.

Accepted Answer

The 95% confidence interval is calculated using the formula: sample mean ± (Z * (population standard deviation / sqrt(sample size))). With a Z value of 1.96 for 95% confidence, the interval is 9 ± (1.96 * (1.6 / 8)), which simplifies to 8.608 to 9.392.

Question 20

A sample of 100 information systems managers had an average hourly income of $40.00 with a standard deviation of $8.00. The 95% confidence interval for the average hourly wage (in $) of all information system managers is

Accepted Answer

To find the 95% confidence interval for the average hourly wage of all information system managers, we will use the formula:

CI = X̄ ± z*(σ/√n)

where X̄ is the sample mean, σ is the population standard deviation, n is the sample size, and z is the z-score corresponding to the desired level of confidence (95% in this case).

Plugging in the given values, we get:

CI = 40 ± 1.96*(8/√100)

CI = 40 ± 1.568

CI = (40 - 1.568, 40 + 1.568)

CI = (38.432, 41.568)

Therefore, the 95% confidence interval for the average hourly wage of all information system managers is (38.432, 41.568). The closest choice to this interval is (C) 38.432 to 41.568.

Quiz 8: Interval Estimation