A sample of 400 items is taken from a population whose standard deviation is 10. The mean of the sample is 40. Test whether the sample has come from a population with mean 38. Also calculate 95% confidence interval for the population.

Well, well, well, looks like we have some statistical fun to dive into! Let's get started.

To test whether the sample has come from a population with a mean of 38, we can use a t-test. The formula for a t-test is:

t = (sample mean - population mean) / (sample standard deviation / √sample size)

In this case, our sample mean is 40, the population mean we want to compare it to is 38, the population standard deviation is 10, and the sample size is 400. Plug those values into the formula, my friend, and you'll get your t-value.

Calculating this, we find that the t-value is 4.0. Now, we have to compare that to the critical value for a t-distribution at a significance level of 0.05. For a two-tailed test like this, the critical values are ±1.96.

Since our t-value of 4.0 is outside the critical value range, we reject the null hypothesis. In other words, we have evidence that the sample did not come from a population with a mean of 38.

Alrighty then, now let's calculate the 95% confidence interval for the population mean. We can use the formula:

Confidence interval = sample mean ± (critical value * standard error)

Where the critical value for a 95% confidence interval with a sample size of 400 is approximately 1.96, and the standard error can be calculated as:

Standard error = population standard deviation / √sample size

So let's do some math! Plugging in our numbers, we find that the standard error is 0.5. Multiplying this by 1.96, we get 0.98.

Now, we can calculate the confidence interval. The sample mean is 40, so the lower bound of the interval will be 40 - 0.98 = 39.02, and the upper bound will be 40 + 0.98 = 40.98.

So, we can say with 95% confidence that the population mean falls within the range of 39.02 to 40.98.

I hope that puts a smile on your face while dealing with statistics! If you have any more questions or need further clarification, feel free to ask!

To test whether the sample comes from a population with a mean of 38, we can perform a hypothesis test. Let's assume that the population mean is 38 and set up the hypothesis:

Null hypothesis (H0): The sample comes from a population with a mean of 38.
Alternative hypothesis (Ha): The sample does not come from a population with a mean of 38.

We can use the z-test for this analysis since we know the population standard deviation. The formula for the z-test is:

z = (sample_mean - population_mean) / (population_standard_deviation / sqrt(sample_size))

In this case:
sample_size = 400
sample_mean = 40
population_mean = 38
population_standard_deviation = 10

Calculating the z-score:

z = (40 - 38) / (10 / √400)
z = 2 / (10 / 20)
z = 2 / 0.5
z = 4

The z-score is 4.

Next, we need to determine the critical value for the 95% confidence level. Since this is a two-tailed test, we need to divide the alpha level by 2. For a 95% confidence level, the alpha level is 0.05, so α/2 = 0.025.

Using a z-table or calculator, we can find the critical z-value for 0.025 in each tail, which is approximately 1.96.

Since the absolute value of the calculated z-value (4) is greater than the critical z-value (1.96), we can reject the null hypothesis.

Conclusion: The sample does not come from a population with a mean of 38.

Now, let's calculate the 95% confidence interval for the population mean. We can use the following formula:

Confidence interval = sample_mean ± (critical_value * standard_error)

The critical value for a 95% confidence level is 1.96 (from the z-table).
The standard error can be calculated as:
standard_error = population_standard_deviation / √sample_size
standard_error = 10 / √400
standard_error = 10 / 20
standard_error = 0.5

Plugging the values into the formula:

Confidence interval = 40 ± (1.96 * 0.5)
Confidence interval = 40 ± 0.98
Confidence interval = (39.02, 40.98)

The 95% confidence interval for the population mean is (39.02, 40.98).

To test whether the sample has come from a population with a mean of 38, we can use a hypothesis test. The null hypothesis (H0) assumes that the sample mean is equal to the population mean, while the alternative hypothesis (Ha) assumes that the sample mean is different from the population mean.

Here are the steps to perform the hypothesis test:

1. Set up the hypotheses:
H0: μ = 38 (sample comes from a population with mean 38)
Ha: μ ≠ 38 (sample does not come from a population with mean 38)

2. Determine the significance level (α). Let's use a significance level of 0.05, which is a common choice.

3. Calculate the standard error (SE) of the sample mean using the population standard deviation and the sample size:
SE = σ / √(n)
where σ is the population standard deviation and n is the sample size.
In this case, σ = 10 and n = 400, so SE = 10 / √(400) = 0.5.

4. Calculate the test statistic (z-score):
z = (x - μ) / SE
where x is the sample mean and μ is the population mean under the null hypothesis.
In this case, x = 40 and μ = 38, so z = (40 - 38) / 0.5 = 4.

5. Determine the critical value(s) for the test statistic. Since we are performing a two-tailed test, we divide the significance level by 2 and find the corresponding values from the standard normal distribution table. For a significance level of 0.05, the critical values are approximately ±1.96.

6. Compare the test statistic to the critical value(s):
- If the test statistic falls within the rejection region (outside the critical values), reject the null hypothesis.
- If the test statistic falls within the acceptance region (inside the critical values), fail to reject the null hypothesis.

In this case, the test statistic of 4 falls outside the critical values of ±1.96. Therefore, we reject the null hypothesis and conclude that there is enough evidence to suggest that the sample does not come from a population with a mean of 38.

To calculate the 95% confidence interval for the population mean, we can use the formula:

CI = x ± z * (SE)
where CI is the confidence interval, x is the sample mean, z is the critical value from the standard normal distribution corresponding to the desired confidence level, and SE is the standard error of the sample mean.

For a 95% confidence interval, the critical value is approximately ±1.96.

Substituting the values into the formula:
CI = 40 ± 1.96 * (0.5)
CI = 40 ± 0.98

Therefore, the 95% confidence interval for the population mean is (39.02, 40.98).

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 your Z score and level of significance.

95% = mean ± 1.96 SEm