Respuesta :
Answer:
The 68% confidence interval for μ is (43648.300, 45351.700)
Step-by-step explanation:
We have been given the following data:
Sample size = n = 64
Sample mean = [tex]\bar{x}[/tex] = 44,500
Sample Standard Deviation = s = 6,800
We have to find 68% confidence interval for the given data. Since, value of Population Standard Deviation is unknown, and we have the value of Sample Standard Deviation, we will use t-distribution to find the required confidence interval.
The formula to calculate the confidence interval is:
[tex](\bar{x}-t_{\frac{\alpha}{2} } \times \frac{s}{\sqrt{n}} , \bar{x}-t_{\frac{\alpha}{2} } \times \frac{s}{\sqrt{n}} )[/tex]
Here,
[tex]t_{\frac{\alpha }{2}}[/tex] is the critical t-value for 68% confidence level and n - 1 = 63 degrees of freedom. From the t-table, this critical value comes out to be t = 1.002
Using the values, we get the confidence interval:
[tex](44500-1.002 \times \frac{6800}{\sqrt{64} }, 44500-1.002 \times \frac{6800}{\sqrt{64} })\\\\ (43648.300, 45351.700)[/tex]
Thus, the 68% confidence interval for μ is (43648.300, 45351.700)
Using the t-distribution, it is found that the 68% confidence interval for μ is (43647.96, 45352.04).
We are given the standard deviation for the sample, which is why the t-distribution is used to solve this question.
The information given is:
- Sample mean of [tex]\mu = 44500[/tex].
- Sample standard deviation of [tex]s = 6800[/tex].
- Sample size of [tex]n = 64[/tex].
The confidence interval is:
[tex]\mu \pm t\frac{s}{\sqrt{n}}[/tex]
The critical value, using a t-distribution calculator, for a two-tailed 68% confidence interval, with 64 - 1 = 63 df, is t = 1.0024.
Then, the interval is:
[tex]\mu - t\frac{s}{\sqrt{n}} = 44500 - 1.0024\frac{6800}{\sqrt{64}} = 43647.96[/tex]
[tex]\mu + t\frac{s}{\sqrt{n}} = 44500 + 1.0024\frac{6800}{\sqrt{64}} = 45352.04[/tex]
The 68% confidence interval for μ is (43647.96, 45352.04).
A similar problem is given at https://brainly.com/question/15180581
