Answer:
To find the value of x and the mean length of the lizards, we need to use the properties of median and mean.
(a) To find the value of x, we first need to arrange the lengths of the lizards in ascending order:
17 cm, 20 cm, 23 cm, 28 cm, 30 cm, x cm
Since there are six lizards, the median length is the average of the two middle values. We know that the median length is 25 cm, so:
(23 cm + 28 cm) / 2 = 25 cm
Therefore, we can conclude that:
23 cm < x < 28 cm
To narrow down the range of possible values for x, we can use another property of median: if we replace any value in a data set with a larger or smaller value, the median will also increase or decrease accordingly.
Since we know that the median length is 25 cm and there are three values smaller than it (17 cm, 20 cm, and 23 cm), we can deduce that any value smaller than or equal to 23 cm will result in a median length smaller than 25 cm. Therefore:
x > 23 cm
Similarly, since there are also three values larger than the median (28 cm, 30 cm, and x), any value larger than or equal to 30 cm will result in a median length larger than 25 cm. Therefore:
x < 30 cm
Combining these inequalities gives us:
23 cm < x < 30 cm
Therefore, x can be any value between 23 and 30 centimeters that satisfies this inequality.
(b) To find the mean length of the lizards, we can use the formula:
mean = (sum of all values) / (number of values)
The sum of all lengths is:
17 + 20 + 23 + 28 + 30 + x = 118 + x
Since there are six lizards, the number of values is 6. Therefore:
mean = (118 + x) / 6
To find the mean length, we need to substitute x with its possible values between 23 and 30 cm:
When x = 23 cm: mean = (118 + 23) / 6 = 20.17 cm
When x = 24 cm: mean = (118 + 24) / 6 = 20.37 cm
When x = 25 cm: mean = (118 + 25) / 6 = 20.50 cm
When x = 26 cm: mean = (118 + 26) / 6 = 20.67 cm
When x = 27 cm: mean = (118 + 27) / 6 = 20.83 cm
When x = 28 cm: mean = (118 + 28) / 6 = 21.00 cm
When x = 29 cm: mean = (118 + 29) / 6 = 21.17 cm
When x = 30 cm: mean = (118 + 30) / 6 = 21.33 cm
Therefore, the mean length of the lizards ranges from approximately 20.17 to 21.33 centimeters, depending on the value of x.