The upper limit for a 95% confidence interval is $97.35. Since this is below $100.00, the family can afford to buy the car.
The Breakdown
To calculate the upper limit for a 95% confidence interval, we can use the formula:
Upper limit = mean + (critical value × standard error)
Calculating the standard error, which is the standard deviation divided by the square root of the sample size:
Standard error = standard deviation / sqrt(sample size)
the mean is $94.00,
the standard deviation is $10.00,
the sample size is 36.
Let's calculate the standard error:
Standard error = $10.00 / sqrt(36) = $10.00 / 6 = $1.67
Next, we need to find the critical value for a 95% confidence level. For a sample size of 36, we can use a t-distribution with (n-1) degrees of freedom. The degrees of freedom in this case would be 36 - 1 = 35.
Using a t-table or a statistical software, we can find the critical value for a 95% confidence level with 35 degrees of freedom. Let's assume the critical value is denoted as "t*".
Finally, we can calculate the upper limit:
Upper limit = $94.00 + (t* * $1.67)
Since the upper limit needs to be below $100.00 for the family to afford the car, we can rearrange the equation to solve for t*:
t* = (Upper limit - $94.00) / $1.67
Let's assume the upper limit is $100.00. Plug in the values:
t* = ($100.00 - $94.00) / $1.67 ≈ 3.59
Now, using the t-distribution table or a statistical software, find the critical value for a 95% confidence level with 35 degrees of freedom that is closest to 3.59. Let's assume the critical value is 2.03.
Finally, calculate the upper limit:
Upper limit = $94.00 + (2.03 × $1.67) ≈ $97.35
Therefore, the upper limit for a 95% confidence interval is $97.35. Since this is below $100.00, the family can afford to buy the car.