Question

Random sample of 50 people with a score of 765 with standard deviation ofn98
Can we conclude that the mean score is greater than 720?
Use a=0.01

Answer 1

What is the p value

Answer 2

To determine if we can conclude that the mean score is greater than 720, we can conduct a hypothesis test. We'll be using a one-sample \( z \)-test for this purpose.

### Step 1: Set up the Null and Alternative Hypotheses

- Null Hypothesis (\( H_0 \)): \( \mu \leq 720 \) (the mean score is less than or equal to 720)
- Alternative Hypothesis (\( H_1 \)): \( \mu > 720 \) (the mean score is greater than 720)

### Step 2: Determine the Significance Level

We are using a significance level \( \alpha = 0.01 \).

### Step 3: Calculate the Test Statistic

Using the formula for the \( z \)-test statistic:

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \( \bar{x} = 765 \) (sample mean)
- \( \mu_0 = 720 \) (hypothesized population mean)
- \( \sigma = 98 \) (population standard deviation)
- \( n = 50 \) (sample size)

Now, substituting in the values:

\[
z = \frac{765 - 720}{\frac{98}{\sqrt{50}}}
\]
\[
= \frac{45}{\frac{98}{7.0711}} \quad \text{(since } \sqrt{50} \approx 7.0711\text{)}
\]
\[
= \frac{45}{13.8576} \quad \text{(calculating } \frac{98}{7.0711} \text{)}
\]
\[
= 3.244
\]

### Step 4: Find the Critical Value

For a one-tailed test with \( \alpha = 0.01 \), we look up the critical value \( z_{\alpha} \) from the standard normal distribution table:

\( z_{0.01} \approx 2.33 \)

### Step 5: Make the Decision

Now we compare the calculated \( z \)-value with the critical value:

- Calculated \( z \approx 3.244 \)
- Critical \( z = 2.33 \)

Since \( 3.244 > 2.33 \), we reject the null hypothesis.

### Conclusion

There is sufficient evidence at the \( \alpha = 0.01 \) significance level to conclude that the mean score is greater than 720.

Answer 3

To calculate the \( p \)-value associated with the computed \( z \)-score of \( 3.244 \), we can refer to the standard normal distribution.

### Step 1: Calculate the \( p \)-value

Since we are conducting a one-tailed test, the \( p \)-value is the probability of obtaining a \( z \)-score greater than \( 3.244 \).

We can find this value using a standard normal distribution table, software, or an online calculator. For \( z = 3.244 \):

1. Look up the cumulative probability for \( z = 3.244 \).
2. The cumulative probability, \( P(Z < 3.244) \), gives us the area to the left of \( 3.244 \).
3. The \( p \)-value will then be \( P(Z > 3.244) = 1 - P(Z < 3.244) \).

### Step 2: Using a standard normal table or calculator

Using a calculator or software, we find:

\[
P(Z < 3.244) \approx 0.9994
\]

Now, we calculate the \( p \)-value:

\[
p\text{-value} = 1 - P(Z < 3.244) \approx 1 - 0.9994 = 0.0006
\]

### Conclusion

The \( p \)-value is approximately **0.0006**.

Given that this \( p \)-value is much less than the significance level of \( \alpha = 0.01 \), it provides strong evidence against the null hypothesis, confirming our previous conclusion that the mean score is greater than 720.