Question

the weights for newborn babies is approximately normally distributed with a mean of 6.7 pounds and a standard deviation of 1.6 pounds. consider a group of 800 newborn babies: 1. how many would you expect to weigh between 4 and 8 pounds? 2. how many would you expect to weigh less than 7 pounds? 3. how many would you expect to weigh more than 5 pounds? 4. how many would you expect to weigh between 6.7 and 9 pounds? hint: do not round until you get your final answer. question help: video read message instructor submit question question 27 a set of test scores is normally distributed with a mean of 76 points and a standard deviation of 8 points. round your answers to the nearest percentile. a student with a score of 76 points would be in the th percentile. a student with a score of 68 points would be in the th percentile. a student with a score of 95 points would be in the th percentile.

Explanation:

Newborn Babies Weight (Normal Distribution, $\mu = 6.7$, $\sigma = 1.6$, $n = 800$)

1. Between 4 and 8 pounds

Step1: Calculate z - scores

For $x = 4$: $z_1=\frac{4 - 6.7}{1.6}=\frac{-2.7}{1.6}\approx - 1.6875$
For $x = 8$: $z_2=\frac{8 - 6.7}{1.6}=\frac{1.3}{1.6}=0.8125$

Step2: Find probabilities from z - table

$P(-1.6875 < Z < 0.8125)=P(Z < 0.8125)-P(Z < - 1.6875)$
From z - table: $P(Z < 0.8125)\approx0.7910$, $P(Z < - 1.6875)\approx0.0455$
So, $P\approx0.7910 - 0.0455 = 0.7455$

Step3: Expected count

$800\times0.7455 = 596.4\approx596$

2. Less than 7 pounds

Step1: Calculate z - score

$z=\frac{7 - 6.7}{1.6}=\frac{0.3}{1.6}=0.1875$

Step2: Find probability from z - table

$P(Z < 0.1875)\approx0.5745$

Step3: Expected count

$800\times0.5745 = 459.6\approx460$

3. More than 5 pounds

Step1: Calculate z - score

$z=\frac{5 - 6.7}{1.6}=\frac{-1.7}{1.6}\approx - 1.0625$

Step2: Find probability from z - table

$P(Z > - 1.0625)=1 - P(Z < - 1.0625)$
From z - table: $P(Z < - 1.0625)\approx0.1446$
So, $P\approx1 - 0.1446 = 0.8554$

Step3: Expected count

$800\times0.8554 = 684.32\approx684$

4. Between 6.7 and 9 pounds

Step1: Calculate z - scores

For $x = 6.7$: $z_1 = 0$ (since $\mu = 6.7$)
For $x = 9$: $z_2=\frac{9 - 6.7}{1.6}=\frac{2.3}{1.6}=1.4375$

Step2: Find probability from z - table

$P(0 < Z < 1.4375)=P(Z < 1.4375)-P(Z < 0)$
From z - table: $P(Z < 1.4375)\approx0.9247$, $P(Z < 0)=0.5$
So, $P\approx0.9247 - 0.5 = 0.4247$

Step3: Expected count

$800\times0.4247 = 339.76\approx340$

Test Scores (Normal Distribution, $\mu = 76$, $\sigma = 8$)

1. Score of 76 points

Step1: z - score

$z=\frac{76 - 76}{8}=0$

Step2: Percentile

$P(Z < 0)=0.5$, so 50th percentile

2. Score of 68 points

Step1: z - score

$z=\frac{68 - 76}{8}=\frac{-8}{8}=-1$

Step2: Percentile

From z - table, $P(Z < - 1)\approx0.1587$, so 16th percentile (rounded)

3. Score of 95 points

Step1: z - score

$z=\frac{95 - 76}{8}=\frac{19}{8}=2.375$

Step2: Percentile

From z - table, $P(Z < 2.375)\approx0.9912$, so 99th percentile (rounded)

Final Answers

Newborn Babies:

1. $\boldsymbol{596}$
2. $\boldsymbol{460}$
3. $\boldsymbol{684}$
4. $\boldsymbol{340}$

Test Scores:

• 76 points: $\boldsymbol{50}$th percentile
• 68 points: $\boldsymbol{16}$th percentile
• 95 points: $\boldsymbol{99}$th percentile

Answer:

Newborn Babies Weight (Normal Distribution, $\mu = 6.7$, $\sigma = 1.6$, $n = 800$)

1. Between 4 and 8 pounds

Step1: Calculate z - scores

For $x = 4$: $z_1=\frac{4 - 6.7}{1.6}=\frac{-2.7}{1.6}\approx - 1.6875$
For $x = 8$: $z_2=\frac{8 - 6.7}{1.6}=\frac{1.3}{1.6}=0.8125$

Step2: Find probabilities from z - table

$P(-1.6875 < Z < 0.8125)=P(Z < 0.8125)-P(Z < - 1.6875)$
From z - table: $P(Z < 0.8125)\approx0.7910$, $P(Z < - 1.6875)\approx0.0455$
So, $P\approx0.7910 - 0.0455 = 0.7455$

Step3: Expected count

$800\times0.7455 = 596.4\approx596$

2. Less than 7 pounds

Step1: Calculate z - score

$z=\frac{7 - 6.7}{1.6}=\frac{0.3}{1.6}=0.1875$

Step2: Find probability from z - table

$P(Z < 0.1875)\approx0.5745$

Step3: Expected count

$800\times0.5745 = 459.6\approx460$

3. More than 5 pounds

Step1: Calculate z - score

$z=\frac{5 - 6.7}{1.6}=\frac{-1.7}{1.6}\approx - 1.0625$

Step2: Find probability from z - table

$P(Z > - 1.0625)=1 - P(Z < - 1.0625)$
From z - table: $P(Z < - 1.0625)\approx0.1446$
So, $P\approx1 - 0.1446 = 0.8554$

Step3: Expected count

$800\times0.8554 = 684.32\approx684$

4. Between 6.7 and 9 pounds

Step1: Calculate z - scores

For $x = 6.7$: $z_1 = 0$ (since $\mu = 6.7$)
For $x = 9$: $z_2=\frac{9 - 6.7}{1.6}=\frac{2.3}{1.6}=1.4375$

Step2: Find probability from z - table

$P(0 < Z < 1.4375)=P(Z < 1.4375)-P(Z < 0)$
From z - table: $P(Z < 1.4375)\approx0.9247$, $P(Z < 0)=0.5$
So, $P\approx0.9247 - 0.5 = 0.4247$

Step3: Expected count

$800\times0.4247 = 339.76\approx340$

Test Scores (Normal Distribution, $\mu = 76$, $\sigma = 8$)

1. Score of 76 points

Step1: z - score

$z=\frac{76 - 76}{8}=0$

Step2: Percentile

$P(Z < 0)=0.5$, so 50th percentile

2. Score of 68 points

Step1: z - score

$z=\frac{68 - 76}{8}=\frac{-8}{8}=-1$

Step2: Percentile

From z - table, $P(Z < - 1)\approx0.1587$, so 16th percentile (rounded)

3. Score of 95 points

Step1: z - score

$z=\frac{95 - 76}{8}=\frac{19}{8}=2.375$

Step2: Percentile

From z - table, $P(Z < 2.375)\approx0.9912$, so 99th percentile (rounded)

Final Answers

Newborn Babies:

1. $\boldsymbol{596}$
2. $\boldsymbol{460}$
3. $\boldsymbol{684}$
4. $\boldsymbol{340}$

Test Scores:

• 76 points: $\boldsymbol{50}$th percentile
• 68 points: $\boldsymbol{16}$th percentile
• 95 points: $\boldsymbol{99}$th percentile