Questions: George measured the weight of a random sample of 49 cartons of apples. The mean weight was 45.5 pounds, with a standard deviation of 3. z = (x̄ - μ) / (σ / sqrt(n)) To see if the cartons have a significantly different mean weight from 46 pounds, what would the value of the z-test statistic be? Answer choices are rounded to the hundredths place. 0.13 1.17 -1.17 -0.13

George measured the weight of a random sample of 49 cartons of apples. The mean weight was 45.5 pounds, with a standard deviation of 3. z = (x̄ - μ) / (σ / sqrt(n)) To see if the cartons have a significantly different mean weight from 46 pounds, what would the value of the z-test statistic be? Answer choices are rounded to the hundredths place. 0.13 1.17 -1.17 -0.13
Transcript text: George measured the weight of a random sample of 49 cartons of apples. The mean weight was 45.5 pounds, with a standard deviation of 3 . \[ z=\frac{\bar{x}-\mu}{\sigma / \sqrt{n}} \] To see if the cartons have a significantly different mean weight from 46 pounds, what would the value of the z-test statistic be? Answer choices are rounded to the hundredths place. 0.13 1.17 $-1.17$ $-0.13$
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Solution

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Solution Steps

Step 1: Calculate the Standard Error of the Mean (SEM)

To calculate the SEM, we use the formula \(\text{SEM} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. Given \(\sigma = 3\) and \(n = 49\), we find: \[\text{SEM} = \frac{3}{\sqrt{49}} = 0.43\]

Step 2: Compute the z-test Statistic

Using the formula \(z = \frac{\bar{x} - \mu}{\text{SEM}}\), where \(\bar{x}\) is the sample mean and \(\mu\) is the population mean, we substitute \(\bar{x} = 45.5\), \(\mu = 46\), and the calculated SEM to find: \[z = \frac{45.5 - 46}{0.43} = -1.17\]

Final Answer:

The z-test statistic is -1.17, which indicates how many standard errors the sample mean is from the population mean. This value can be used to determine the significance of the difference between the sample mean and the population mean by referring to the z-table.

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