Question 1172748
Absolutely! Let's break down this problem step by step.

**Understanding X-bar Control Charts**

An X-bar control chart monitors the mean of a process. The control limits (control.png and control2.png, which we'll refer to as UCL and LCL) are typically set at ±3 standard deviations from the process mean.

* **When the process is under control:** This means the process mean is stable, and the sample means should fall within the control limits with a high probability.

**Key Concepts**

* **Normal Distribution:** We assume the sample means follow a normal distribution.
* **Probability within Control Limits:** For a process under control, the probability that a single sample mean falls within the control limits is very high (close to 1). Specifically, within 3 standard deviations it is approximately 0.9973.

**Solving the Problem**

**(a) Probability of Five Consecutive Sample Means Within Limits**

1.  **Probability for one sample mean:**
    * Let P(in) be the probability that a single sample mean falls within the control limits.
    * Since the process is under control, P(in) ≈ 0.9973 (for 3 sigma control limits).

2.  **Probability for five consecutive sample means:**
    * Since the samples are independent, we multiply the probabilities:
        * P(5 consecutive in) = P(in) * P(in) * P(in) * P(in) * P(in) = P(in)^5
        * P(5 consecutive in) = (0.9973)^5 ≈ 0.9866

    * Therefore, the probability that five consecutive sample means stay within the limits is approximately 0.9866.

**(b) Probability of 100 Days of Means Within Limits**

1.  **Probability for 100 days:**
    * Similarly, for 100 days, we raise the probability for one day to the power of 100.
        * P(100 consecutive in) = P(in)^100
        * P(100 consecutive in) = (0.9973)^100 ≈ 0.7631

    * Therefore, the probability that all of the means for 100 days fall within the control limits is approximately 0.7631.

**Summary**

* (a) The probability that five consecutive sample means stay within the limits is approximately 0.9866.
* (b) The probability that all of the means for 100 days fall within the control limits is approximately 0.7631.