Final answer:
To find the 95% confidence interval for any single day's travel time to work based on the recorded times, we calculate the sample mean and estimate the sample standard deviation. We then use the t-distribution to determine the margin of error and calculate the interval. Without the actual standard deviation value, we suggest the widest range option (c) 34 to 60 minutes, which would likely encompass the mean and allow for natural variation, but we cannot be certain.
Step-by-step explanation:
To determine a 95% confidence interval for a single trip's travel time based on the 10 recorded times, we must calculate the sample mean and standard deviation. Then we can use these to find the interval that contains the central 95% of the data assuming a normal distribution.
The data set is: 35, 49, 51, 37, 59, 38, 41, 34, 39, 45. To start, let's find the mean and standard deviation:
- Mean (average) = (35+49+51+37+59+38+41+34+39+45) / 10 = 428 / 10 = 42.8 minutes
- Standard Deviation (s) is calculated using the formula for a sample standard deviation. We'll skip the full calculation here for brevity.
For a 95% confidence interval of a single observation from a normally distributed data set, we use the formula:
Single observation 95% CI = Mean ± (t-value × (s/√n))
Where t-value is based on t-distribution with n-1 degrees of freedom (for 10 observations, n-1 = 9). For 95% confidence, the t-value corresponding to 9 degrees of freedom is approximately 2.262 (this may vary depending on the source of the t-table).
Calculating the confidence interval:
- We estimate the standard deviation s (using the data provided) and then divide it by the square root of the number of observations (√n) to get the standard error.
- Then we multiply the standard error by the t-value to find our margin of error.
- Finally, we add and subtract this margin of error from the mean to get the confidence interval.
Given that this is a theoretical calculation and we don't have an actual standard deviation, we are unable to calculate the exact confidence interval here. However, based on the options given and the mean of 42.8 minutes, we can suggest an answer.
Looking at the answer choices, the widest interval that would likely encompass the mean and allow for natural variation in the data would be option (c) 34 to 60 minutes. However, without the standard deviation, we cannot be certain of this answer.