Last year Marion Dairies decided to enter the yogurt market, and it began cautiously by producing, distributing, and marketing a single flavor - a blueberry-flavored yogurt that it calls Blugert. The company' initial venture into the yogurt market has been very successful; sales of Blugert are higher than expected, and consumers' ratings of the product have a mean of 80 and a standard deviation of 25 on a 100-point scale for which 100 is the most favorable score and zero is the least favorable score. Past experience has also shown Marion Dairies that a consumer who rates one of its products with a score greater than 75 on this scale will consider purchasing the product, and a score of 75 or less indicates that the consumer will not consider purchasing the product. Emboldened by the success and popularity of its blueberry-flavored yogurt, Marion Dairies management is now considering the introduction of a second flavor. Marion's marketing department is pressing to extend the product line through the introduction of a strawberryflavored yogurt that would be called Strawgurt, but senior managers are concerned about whether or not Strawgurt will increase Marion's market share by appealing to potential customers who do not like Blugert. That is, the goal in offering the new product is to increase the market share rather than cannibalize existing sales of Blugert. The marketing department has proposed giving tastes of both Blugert and Strawgurt to a simple random sample of 50 customers and asking each of them to rate the two yogurts on the 100-point scale. If the mean score given to Blugert by this sample of consumers is 75 or less, Marion's senior management believes the sample can be used to assess whether Strawgurt will appeal to potential customers who do not like Blugert. 1. Calculate the probability that the mean score of Blugert given by the simple random sample of Marion Dairies customers will be 75 or less. 2. If the Marketing Department increases the sample size to 150, what is the probability that the mean score of Blugert given by the simple random sample of Marion Dairies customers will be 75 or less? 3. Explain to Marion Dairies senior management why the probability that the mean score of Blugert given by the simple random sample of Marion Dairies customers will be 75 or less differs for these two sample sizes.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Last year Marion Dairies decided to enter the yogurt market, and it began cautiously by
producing, distributing, and marketing a single flavor - a blueberry-flavored yogurt that it calls
Blugert. The company' initial venture into the yogurt market has been very successful; sales of
Blugert are higher than expected, and consumers' ratings of the product have a mean of 80 and
a standard deviation of 25 on a 100-point scale for which 100 is the most favorable score and
zero is the least favorable score. Past experience has also shown Marion Dairies that a
consumer who rates one of its products with a score greater than 75 on this scale will consider
purchasing the product, and a score of 75 or less indicates that the consumer will not consider
purchasing the product.
Emboldened by the success and popularity of its blueberry-flavored yogurt, Marion Dairies
management is now considering the introduction of a second flavor. Marion's marketing
department is pressing to extend the product line through the introduction of a strawberryflavored
yogurt that would be called Strawgurt, but senior managers are concerned about
whether or not Strawgurt will increase Marion's market share by appealing to potential
customers who do not like Blugert. That is, the goal in offering the new product is to increase
the market share rather than cannibalize existing sales of Blugert. The marketing department
has proposed giving tastes of both Blugert and Strawgurt to a simple random sample of 50
customers and asking each of them to rate the two yogurts on the 100-point scale. If the mean
score given to Blugert by this sample of consumers is 75 or less, Marion's senior management
believes the sample can be used to assess whether Strawgurt will appeal to potential customers
who do not like Blugert.
1. Calculate the
sample of Marion Dairies customers will be 75 or less.
2. If the Marketing Department increases the
that the mean score of Blugert given by the simple random sample of Marion Dairies
customers will be 75 or less?
3. Explain to Marion Dairies senior management why the probability that the mean score
of Blugert given by the simple random sample of Marion Dairies customers will be 75 or
less differs for these two sample sizes.
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