Question

The National Coffee Association reported that 62% of U.S. adults drink coffee daily. A random sample of 225 U.S. adults is selected. Round your answers to four decimal places as needed. Part 1 Find the mean μ The mean μ is Part 2 Find the standard deviation σ The standard deviation σ is Part 3 Find the probability that more than 67% of the sampled adults drink coffee daily The probability that more than 67% of the sampled adults drink coffee daily is Part 4 Find the probability that the proportion of the sampled adults who drink coffee daily is between 0.55 and 0.67. The probability that the proportion of the sampled adults who drink coffee daily is between 0.55 and 0.67 is Part 5 Find the probability that less than 59% of sampled adults drink coffee daily. The probability that less than 59% of sampled adults drink coffee daily is Part 6 Would it be unusual if less than 54% of the sampled adults drink coffee daily? It (select) probability is be unusual if less than 54% of the sampled adults drink coffee daily, since the

Answer 1

The mean and standard deviation are calculated from the given data. Probability values for different scenarios are calculated using the normal distribution.

Step-by-step explanation:

Part 1:

To find the mean, we multiply the percentage by the sample size:

Mean (μ) = 0.62 * 225 = 139.5

Part 2:

To find the standard deviation, we use the formula:

Standard Deviation (σ) = sqrt(n * p * (1 - p))

where n is the sample size and p is the percentage. Plugging in the values, we get:

Standard Deviation (σ) = sqrt(225 * 0.62 * (1 - 0.62)) = 8.2221

Part 3:

To find the probability that more than 67% of the sampled adults drink coffee daily, we can use the normal distribution. The z-score can be calculated using the formula:

z = (x - μ) / σ

where x is the value we are interested in, μ is the mean, and σ is the standard deviation. Plugging in the values, we get:

z = (0.67 - 0.62) / (8.2221 / sqrt(225)) = 0.5596

Using a z-table or calculator, we can find the probability corresponding to the z-score. The probability of more than 67% of the sampled adults drinking coffee daily is approximately 0.2881.

Part 4:

To find the probability that the proportion of sampled adults who drink coffee daily is between 0.55 and 0.67, we can use the normal distribution. We calculate the z-scores for the lower and upper bounds:

z1 = (0.55 - 0.62) / (8.2221 / sqrt(225)) = -0.3427

z2 = (0.67 - 0.62) / (8.2221 / sqrt(225)) = 0.5596

Using a z-table or calculator, we can find the probabilities corresponding to the z-scores. The probability that the proportion is between 0.55 and 0.67 is the difference between these two probabilities, which is approximately 0.4412.

Part 5:

To find the probability that less than 59% of sampled adults drink coffee daily, we can use the normal distribution. The z-score can be calculated using the formula:

z = (x - μ) / σ

Plugging in the values, we get:

z = (0.59 - 0.62) / (8.2221 / sqrt(225)) = -0.1712

Using a z-table or calculator, we can find the probability corresponding to the z-score. The probability of less than 59% of sampled adults drinking coffee daily is approximately 0.4332.

Part 6:

To determine if it would be unusual for less than 54% of the sampled adults to drink coffee daily, we need to compare the probability to a significance level. If the probability is less than the significance level (usually 0.05), then it would be considered unusual. However, we don't have the probability value for this case, so we cannot determine if it would be unusual or not.