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The average price of a gallon of unleaded regular gasoline was reported to be $2.34 in northern Kentucky (The Cincinnati Enquirer, January 21, 2006). Use this price as the population mean, and assume the population standard deviation is $.20.
1. What is the probability that the mean price for a sample of 30 service stations is within $.03 of the population mean (to 4 decimals)?
2. What is the probability that the mean price for a sample of 50 service stations is within $.03 of the population mean (to 4 decimals)?
3. What is the probability that the mean price for a sample of 100 service stations is within $.03 of the population mean (to 4 decimals)?
4. Calculate the sample size necessary to guarantee at least .95 probability that the sample mean is within $.03 of the population mean (0 decimals).
# What is the probability that the mean price for a sample of 30 service stations is within $.03 of the population mean (to 4 decimals)?
.5878
# What is the probability that the mean price for a sample of 50 service stations is within $.03 of the population mean (to 4 decimals)?
.7108
# What is the probability that the mean price for a sample of 100 service stations is within $.03 of the population mean (to 4 decimals)?
.8664
# Calculate the sample size necessary to guarantee at least .95 probability that the sample mean is within $.03 of the population mean (0 decimals).
?
What is the probability that the mean price for a sample of 30 service stations is within $.03 of the population mean (to 4 decimals)?
.5878
# What is the probability that the mean price for a sample of 50 service stations is within $.03 of the population mean (to 4 decimals)?
.7108
# What is the probability that the mean price for a sample of 100 service stations is within $.03 of the population mean (to 4 decimals)?
.8664
# Calculate the sample size necessary to guarantee at least .95 probability that the sample mean is within $.03 of the population mean (0 decimals).
?
To answer these questions, we will use the normal distribution and the properties of the sampling distribution of the mean.
1. Probability for a sample of 30 service stations:
To find the probability that the mean price for a sample of 30 service stations is within $0.03 of the population mean, we need to calculate the z-score for a $0.03 interval and then find the corresponding probability.
First, we calculate the standard error, which is the population standard deviation divided by the square root of the sample size:
Standard Error (SE) = $0.20 / √30 ≈ $0.03651
Next, we calculate the z-score using the formula:
z = (Sample Mean - Population Mean) / Standard Error
The lower z-score bound is:
Lower Z-score = ($2.34 - $2.34 - $0.03) / $0.03651
The upper z-score bound is:
Upper Z-score = ($2.34 + $0.03 - $2.34) / $0.03651
Now, we can use a standard normal distribution table or a statistical calculator to find the probability associated with the corresponding z-scores. The probability would be the difference between the cumulative probabilities associated with the upper and lower z-scores.
2. Probability for a sample of 50 service stations:
The process is similar to the previous question, but the sample size changes. Calculate the standard error and then find the corresponding z-scores. Finally, find the probability using either a standard normal distribution table or a statistical calculator.
3. Probability for a sample of 100 service stations:
Again, the process is the same as in the previous two questions. Calculate the standard error, find the z-scores, and calculate the probability.
4. Sample size necessary for at least 0.95 probability:
To find the minimum sample size needed to guarantee a probability of at least 0.95 (95%) that the sample mean is within $0.03 of the population mean, we can use the formula for sample size determination:
Sample Size = (Z-score * Population Standard Deviation / Margin of Error)^2
Since we want at least 95% confidence, the Z-score corresponding to a 95% confidence level (alpha level of 0.05) is approximately 1.96.
Sample Size = (1.96 * $0.20 / $0.03)^2
By substituting the values into the formula, you can calculate the required sample size to guarantee the desired probability.
Remember to use an appropriate statistical calculator or table to find the z-scores and their associated probabilities accurately.