A 4 to 7 month old baby sleeps an average of 14 hours per day. At this age, babies sleep patterns follow normal distribution with the standard deviation of 1.6
a) is it unusual for a baby to sleep under 12 hours a day?
b) In a random sample of 10 babies of this age,find the probability that the mean number of hours of sleep is below 13 hours.
To answer these questions, we will use the concept of z-scores and the properties of the normal distribution.
a) To determine if it is unusual for a baby to sleep under 12 hours a day, we need to calculate the z-score for this value. The z-score tells us how many standard deviations an observation is away from the mean.
To calculate the z-score, we use the formula: z = (x - μ) / σ, where x is the observed value, μ is the mean, and σ is the standard deviation.
In this case, the observed value is 12 hours, the mean is 14 hours, and the standard deviation is 1.6. Plugging these values into the formula, we get: z = (12 - 14) / 1.6 = -1.25.
Now, we can look up the z-score in a standard normal distribution table or use a calculator to find the corresponding p-value. A p-value less than 0.05 (or 5%) is generally considered rare or unusual.
b) To find the probability that the mean number of hours of sleep is below 13 hours in a random sample of 10 babies, we will use the sampling distribution of the sample mean.
Since we are dealing with a sample mean, we need to use the standard error of the mean (SE) instead of the standard deviation (σ). The formula for SE is σ / sqrt(n), where σ is the standard deviation and n is the sample size.
In this case, the standard deviation is still 1.6, and the sample size is 10. Plugging these values into the formula, we get: SE = 1.6 / sqrt(10) = 0.506.
Next, we need to calculate the z-score using the formula: z = (x - μ) / SE. The observed value (x) is 13 hours, the mean (μ) is still 14 hours, and SE is 0.506. Plugging these values into the formula, we get: z = (13 - 14) / 0.506 = -1.98.
Now, we can look up the z-score in a standard normal distribution table or use a calculator to find the corresponding p-value. The p-value obtained represents the probability that the mean number of hours of sleep is below 13 hours in a random sample of 10 babies.