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QuestionKey Club students at Peconic High made trips to Echo Beach.
a. Total students? b. Find mean, median, mode, range, variance, and standard deviation (round to 1 decimal). c. For a normal distribution with mean 6.2 and SD 2, what % went >8 times? (round to nearest %) d. What % went 5-7 times? (round to nearest %)

Studdy Solution

STEP 1

Assumptions1. The table shows the number of times each student in the Key Club went to Echo Beach last summer. . The frequency, ff, represents the number of students who made the corresponding number of trips.
3. The distribution of the number of visits per person last summer is normal with mean6. and standard deviation.

STEP 2

First, we need to find the total number of students in the Key Club. We can do this by adding up the frequencies.
Total students=∑fTotal\, students = \sum f

STEP 3

Now, plug in the given frequencies to calculate the total number of students.
Total students=8+7+11+15+3+Total\, students =8 +7 +11 +15 +3 +

STEP 4

Calculate the total number of students.
Total students=8+7+11+15+3+4=48Total\, students =8 +7 +11 +15 +3 +4 =48

STEP 5

Next, we need to find the mean of the distribution. The mean is the sum of the product of each value and its frequency, divided by the total frequency.
Mean=∑(value×frequency)Total frequencyMean = \frac{\sum (value \times frequency)}{Total\, frequency}

STEP 6

Now, plug in the given values and frequencies to calculate the mean.
Mean=(5×8)+(6×)+(8×11)+(9×15)+(12×3)+(14×4)48Mean = \frac{(5 \times8) + (6 \times) + (8 \times11) + (9 \times15) + (12 \times3) + (14 \times4)}{48}

STEP 7

Calculate the mean.
Mean=(5×)+(6×7)+(×11)+(9×15)+(12×3)+(14×4)48=7.9Mean = \frac{(5 \times) + (6 \times7) + ( \times11) + (9 \times15) + (12 \times3) + (14 \times4)}{48} =7.9

STEP 8

To find the median, we need to arrange the data in ascending order and find the middle value. If the total number of data points is even, the median is the average of the two middle values.
Since we have48 students, the median will be the average of the24th and25th values.

STEP 9

Looking at the cumulative frequency, we see that the24th and25th students both made8 trips. So, the median is8.

STEP 10

The mode is the value with the highest frequency. Looking at the table, we see that the mode is9, since it has the highest frequency of15.

STEP 11

The range is the difference between the highest and lowest values. In this case, the range is 14−5=914 -5 =9.

STEP 12

The variance is the average of the squared differences from the mean. We calculate it as followsVariance=∑[(value−mean)2×frequency]Total frequencyVariance = \frac{\sum [(value - mean)^2 \times frequency]}{Total\, frequency}

STEP 13

Now, plug in the given values, frequencies, and the calculated mean to calculate the variance.
Variance=(5−7.9)2×8+(6−7.9)2×7+(8−7.9)2×11+(9−7.9)2×15+(12−7.9)2×3+(−7.9)2×48Variance = \frac{(5 -7.9)^2 \times8 + (6 -7.9)^2 \times7 + (8 -7.9)^2 \times11 + (9 -7.9)^2 \times15 + (12 -7.9)^2 \times3 + ( -7.9)^2 \times}{48}

STEP 14

Calculate the variance.
Variance=(−7.9)2×8+(6−7.9)2×7+(8−7.9)2×11+(9−7.9)2×+(12−7.9)2×3+(14−7.9)2×448=6.Variance = \frac{( -7.9)^2 \times8 + (6 -7.9)^2 \times7 + (8 -7.9)^2 \times11 + (9 -7.9)^2 \times + (12 -7.9)^2 \times3 + (14 -7.9)^2 \times4}{48} =6.

STEP 15

The standard deviation is the square root of the variance.
Standard deviation=VarianceStandard\, deviation = \sqrt{Variance}

STEP 16

Now, plug in the calculated variance to find the standard deviation.
Standard deviation=6.Standard\, deviation = \sqrt{6.}

STEP 17

Calculate the standard deviation.
Standard deviation=6.=2.5Standard\, deviation = \sqrt{6.} =2.5

STEP 18

To find the percent of visitors who went to the beach more than eight times, we need to calculate the z-score for8 and then find the area to the right of this z-score on the standard normal distribution.
The z-score is calculated as followsz=value−meanstandard deviationz = \frac{value - mean}{standard\, deviation}

STEP 19

Now, plug in the given value, mean, and standard deviation to calculate the z-score.
z=8−6.z = \frac{8 -6.}{}

STEP 20

Calculate the z-score.
z=8−6.=0.9z = \frac{8 -6.}{} =0.9

STEP 21

We can now use a standard normal distribution table or a calculator to find the area to the right of this z-score, which represents the percent of visitors who went to the beach more than eight times.
The area to the right of z=0.9z =0.9 is approximately0.1841, or18.41%.

STEP 22

To find the percent of visitors who went to the beach between five and seven times, inclusive, we need to calculate the z-scores for5 and7 and then find the area between these z-scores on the standard normal distribution.

STEP 23

Now, plug in the given values, mean, and standard deviation to calculate the z-scores.
z1=5−6.z1 = \frac{5 -6.}{}z=7−6.z = \frac{7 -6.}{}

STEP 24

Calculate the z-scores.
z1=−6.=−0.6z1 = \frac{ -6.}{} = -0.6z=7−6.=0.4z = \frac{7 -6.}{} =0.4

STEP 25

We can now use a standard normal distribution table or a calculator to find the area between these z-scores, which represents the percent of visitors who went to the beach between five and seven times, inclusive.
The area between z=−0.z = -0. and z=0.4z =0.4 is approximately0.3859, or38.59%.

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