Let's solve the problem step-by-step based on the given details:
### Part a
To find [tex]\(\sigma_{\bar{x}}\)[/tex], we use the formula:
[tex]\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Given:
- [tex]\(\sigma = 6.2\)[/tex]
- [tex]\(n = 7\)[/tex]
First, we calculate the denominator:
[tex]\[
\sqrt{7} = 2.6457513110645907
\][/tex]
Then, we divide the population standard deviation by this value:
[tex]\[
\sigma_{\bar{x}} = \frac{6.2}{2.6457513110645907} \approx 2.343
\][/tex]
Rounding to one decimal place:
[tex]\[
\sigma_{\bar{x}} \approx 2.3
\][/tex]
### Part b
We need to find [tex]\(P(\bar{x} < 19)\)[/tex] using the rounded [tex]\(\sigma_{\bar{x}} = 2.3\)[/tex].
First, we calculate the z-score:
[tex]\[
z = \frac{19 - 19.9}{2.3} \approx -0.3913
\][/tex]
Using normal distribution tables or a cumulative distribution function (CDF) calculator, we find the probability corresponding to this z-score:
[tex]\[
P(\bar{x} < 19) \approx 0.3478
\][/tex]
### Part c
Now, we find [tex]\(P(\bar{x} < 19)\)[/tex] using the exact value of [tex]\(\sigma_{\bar{x}}\)[/tex]:
[tex]\[
\sigma_{\bar{x}} = \frac{6.2}{\sqrt{7}} \approx 2.343
\][/tex]
Calculate the z-score again:
[tex]\[
z = \frac{19 - 19.9}{2.343} \approx -0.3847
\][/tex]
Using CDF calculator for the exact z-score, we get:
[tex]\[
P(\bar{x} < 19) \approx 0.3505
\][/tex]
### Part d
We need to find the value of [tex]\(a\)[/tex] such that [tex]\(P(\bar{x} < a) = 0.62\)[/tex].
First, find the z-score for which the cumulative probability is 0.62. Using the Z-table or inverse CDF:
[tex]\[
z \approx 0.305
\][/tex]
Using the exact [tex]\(\sigma_{\bar{x}}\)[/tex], we calculate [tex]\(a\)[/tex]:
[tex]\[
a = 19.9 + 0.305 \times 2.343 \approx 19.9 + 0.714315 \approx 20.62
\][/tex]
Hence, gathering all the results:
- a. [tex]\( \sigma_{\bar{x}} \approx 2.3 \)[/tex]
- b. [tex]\( P(\bar{x} < 19) \approx 0.3478 \)[/tex]
- c. [tex]\( P(\bar{x} < 19) \approx 0.3505 \)[/tex]
- d. [tex]\( a \approx 20.62 \)[/tex]
