A sample of 100 iron bars is said to be drawn from a large number of bars.

Whose lengths are normally distributed with mean 4 feet and S.D 0.6ft. If the
sample mean is 4.2 ft, can the sample be regarded as a truly random sample?

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√n

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.

A sample of 100 iron bars is said to be drawn from a large number of bars.

Whose lengths are normally distributed with mean 4 feet and S.D 0.6ft. If the
sample mean is 4.2 ft, can the sample be regarded as a truly random sample?

Well, iron bars are known for being tough and rigid, but when it comes to randomness, they tend to lack flexibility. So, if the sample mean is 4.2 feet instead of the expected mean of 4 feet, I'd say there's something fishy going on. Perhaps these iron bars have been working out and are trying to impress you with their extra length. In all seriousness, though, based on statistical measures, a random sample should closely resemble the population it's drawn from. Since the sample mean is slightly higher than the expected mean, it indicates that there might be some bias or non-randomness in the sample selection. So, I'd hesitate to regard it as a truly random sample. It's a little like finding a clown in a box of gummy bears – something doesn't quite add up.

To determine if the sample can be regarded as a truly random sample, we can use a hypothesis test.

Step 1: State the null hypothesis and alternative hypothesis.
Null hypothesis (H0): The sample is drawn from a truly random population.
Alternative hypothesis (Ha): The sample is not drawn from a truly random population.

Step 2: Determine the significance level.
Let's assume a significance level (alpha) of 0.05, meaning we would be willing to make a Type I error (rejecting the null hypothesis when it is true) 5% of the time.

Step 3: Calculate the test statistic.
We can use the z-test statistic to test the sample mean against the population mean. The formula for the z-test statistic is:

z = (x̄ - μ) / (σ / sqrt(n))

Where:
x̄ = sample mean
μ = population mean
σ = population standard deviation
n = sample size

Given the following values:
x̄ = 4.2 feet
μ = 4.0 feet
σ = 0.6 feet
n = 100

Let's calculate the z-test statistic:

z = (4.2 - 4.0) / (0.6 / sqrt(100))
z = 0.2 / (0.6 / 10)
z = 0.2 / 0.06
z ≈ 3.33

Step 4: Determine the critical value.
Since we assumed a significance level (alpha) of 0.05, the critical value corresponding to a two-tailed test at alpha/2 = 0.025 is approximately 1.96 (using a standard normal distribution table).

Step 5: Compare the test statistic with the critical value.
Since the test statistic (z = 3.33) is greater than the critical value (1.96), we reject the null hypothesis.

Step 6: Interpret the result.
Based on the analysis, we can conclude that the sample cannot be regarded as a truly random sample.

To determine whether the sample can be regarded as a truly random sample, we can perform a hypothesis test.

The first step is to set up the null and alternative hypotheses. In this case, the null hypothesis (H0) would be that the sample is drawn from a population with a mean of 4 feet, while the alternative hypothesis (Ha) would be that the sample is drawn from a population with a mean different from 4 feet.

Next, we need to choose a significance level, denoted as α, which represents the maximum probability of rejecting the null hypothesis when it is actually true. A common value for α is 0.05, which corresponds to a 5% significance level.

We can then perform a hypothesis test using the sample mean and the information given about the population mean and standard deviation. Here are the steps to carry out the test:

1. Calculate the standard error (SE) of the sample mean using the population standard deviation and the sample size:
SE = (0.6 ft) / √100 = 0.06

2. Calculate the test statistic, which follows a Student's t-distribution since the population standard deviation is unknown:
t = (sample mean - population mean) / SE = (4.2 ft - 4 ft) / 0.06

3. Determine the critical value(s) for the test statistic at the chosen significance level of α. This can be done by looking up the corresponding values in the t-distribution table or using statistical software.

4. Compare the test statistic to the critical value(s):
- If the test statistic falls within the acceptance region (i.e., it is not more extreme than the critical value), we fail to reject the null hypothesis, indicating that the sample can be regarded as a truly random sample.
- If the test statistic falls within the rejection region (i.e., it is more extreme than the critical value), we reject the null hypothesis, suggesting that the sample cannot be regarded as a truly random sample.

By performing the calculations and comparing the test statistic to the critical value(s), you can determine whether the sample can be regarded as a truly random sample.