Question

a) if 40 is added to each of the data values above, what is the new mean? what is the new standard deviation? what is the new variance? e) if each data value above is multiplied by 15, what is the new mean? 59.47×15 = 892.05 what is the new standard deviation? what is the new variance?

Explanation:

Step1: Recall mean - addition property

If \(x_1,x_2,\cdots,x_n\) is a data - set with mean \(\mu\) and we add a constant \(c\) to each data value, the new mean \(\mu_{new}\) is \(\mu + c\). Let the original mean be \(\mu\). When \(c = 10\), the new mean is \(\mu+10\). But we don't know the original mean from the given text. However, when each data - value is multiplied by \(a = 15\), if the original mean is \(\mu\), the new mean \(\mu_{new}=a\mu\). Given \(\mu = 54.67\), \(\mu_{new}=15\times54.67 = 820.05\).

Step2: Recall standard - deviation property

If we add a constant \(c\) to each data value in a data - set, the standard deviation \(\sigma\) remains the same because the spread of the data relative to each other does not change. When we multiply each data value by a constant \(a\), the new standard deviation \(\sigma_{new}=a\sigma\). Since we don't know the original standard deviation \(\sigma\), we can't calculate the new standard deviation when adding 10. When \(a = 15\), if the original standard deviation is \(\sigma\), the new standard deviation is \(15\sigma\).

Step3: Recall variance property

The variance \(s^{2}\) is related to the standard deviation \(s\) by \(s^{2}=\sigma^{2}\). When we add a constant \(c\) to each data value, the variance remains the same. When we multiply each data value by \(a\), the new variance \(s_{new}^{2}=a^{2}s^{2}\). When \(a = 15\), if the original variance is \(s^{2}\), the new variance is \(15^{2}s^{2}=225s^{2}\).

Since we don't know the original standard deviation and variance:

1. When adding 10 to each data value:

• New mean: Original mean + 10 (unknown without original mean)
• New standard deviation: Same as original (unknown)
• New variance: Same as original (unknown)

1. When multiplying each data value by 15:

• New mean: \(820.05\)
• New standard deviation: \(15\times\) original standard deviation (unknown)
• New variance: \(225\times\) original variance (unknown)

Answer:

When adding 10 to each data value:

• New mean: Original mean + 10
• New standard deviation: Same as original
• New variance: Same as original

When multiplying each data value by 15:

• New mean: \(820.05\)
• New standard deviation: \(15\times\) original standard deviation
• New variance: \(225\times\) original variance